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REMARKS 

ON 

MATHEMATICAL OR DEMONSTRATIVE 
REASONING: 

ITS CONNEXION WITH LOGIC; 
AND ITS APPLICATION TO SCIENCE, 

PHYSICAL AND METAPHYSICAL, 
WITH REFERENCE TO SOME RECENT PUBLICATIONS. 

Vld^l EDWARD ^AGART, F.G.S. 

.•.IINISTER OF THE CHAPEL llf LITTLE PORTLAND-STREET, REGENT-STREET. 



^ " The light of human minds is perspicuous words ; but by definition first 
snuffed, and purged from ambiguity." — Hobbes. 

" It is an old remark that geometry is the best logic." — Berkeley's Analyst. 



LONDON: 
JOHN GREEN, 121, NEWGATE STREET. 

1837. 



^0 



1^ 



.Tlil) 1!Y lUCIIAnC KINDER, 
NEW STREET, FETTER LAN 



THE REV. W. TURNER, JUN. 
OF HALIFAX. 



My Dear Sir, 
To your valued instructions in Manchester College, 
York, I am indebted for much of the pleasure which I 
have occasionally derived from metaphysical inquiries 
and discussions. In the metaphysical as well as theo- 
logical department of that Institution, we were taught to 
study with care and candour the best works, not to cavil 
and confute, " nor to believe and take for granted, but 
to weigh and consider." To you, therefore, as one well 
skilled in weighing arguments and detecting fallacies, 
I beg leave to inscribe this little volume, and remain 
Your grateful pupil, 

EDWARD TAGART. 

Bayswater, 
September 1S37. 



a 2 



PREFACE. 



The remarks here offered to the reader 
were commenced some time ago, with the 
design of adapting them to the pages of a 
Heview or Magazine. Hence they have 
perhaps too much of a critical and contro- 
versial air for a distinct Essay on an abstract 
subject. But the remarks grew under my 
hands ; and becoming, as they grew, less and 
less fit for any existing periodical, they are 
now presented, but with great diffidence, in 



Vi PREFACE. 

a separate form — with what advantage, the 
reader of course will judge. 

The well-instructed student of mathema- 
tics, of logic, of the nature and theory of 
language, and of what is called moral evi- 
dence, will be apt to remark at the close of the 
work, that it makes no specific addition to the 
amount of his knowledge. But if the views 
presented even to him be admitted to be 
correct as far as they go ; if some thoughts 
are here conveniently brought together which 
must at least be sought for in widely-scat- 
tered sources ; if they have the good eftect of 
awakening the attention of the less profound 
to important points connected with the sub- 
ject before him, hitherto overlooked ; if they 
suggest or stimulate inquiries worthy of con- 
tinuance, — the publication will not be in vain. 
Much use has been made of the sentiments 



PREFACE. Vll 

of others, so as to form a sort of philo- 
sopliical discussion in which many authors 
are made to speak for themselves. But an 
ample apology for this, if one be necessary, 
will be found in the words of Dr. Law in his 
preface to the translation of Dr. King's Essay 
on the Origin of Evil : — 

" A writer often does more good by show- 
ing the use of some of those many volumes 
which we have already, than by offering new 
ones, though this be of much less advantage 
to his own character. I determined there- 
fore not to say anything myself where I 
could bring another conveniently to say it 
for me ; and transcribed only so much from 
others as was judged absolutely necessary to 
give the reader a short view of the subject, 
and by that sketch to induce those who have 
leisure, opportunity, and inclination to go 



Vlli PREFACE. 

further and consult the originals, and to 
afford some present satisfaction to those who 
have not, 

" But how judiciously this is performed, 
the notes themselves must testify." 



CONTENTS. 



Page. 
Introductor))' remarks ..... 1 

Mathematical reasoning sets out fj'om definitions . 5 
Tliese definitions settle the meaning of terms . 1 

These terms, signs of ideas of figure and quantity — 

ideas originating in sensible impressions . . 20 

Mathematical reasoning supported by diagrams, or 

evidence of the senses . - ... 22 
Mr. Whewell's language on experience as the source 

of mathematical conceptions, criticised . . 27 

Beddoes on Demonstrative Evidence, and Playfair 

on Beddoes, considered . . . . .33 

Influence of habit, or constant connexion of the 

terms with the same ideas, in producing assent to 

mathematical processes . . . . .36 

Fewness of premises and of terms in n:iathematical 

reasoning ....... 38 

Distinctness and simplicity of ideas of number and 

figure 40 

Final and essential characteristic of deinonstrative 

reasoning ....... 44 



CONTENTS. 



SECTION II. 

Page. 

Mr. Dugald Stewart's, Dr. Wlaately's, and the Edin- 
burgh reviewer's remarks on mathematical and 
general reasoning contrasted . . . .50 
Sir John Herschel's character of Sir Isaac New- 
ton 57 

Whately's opinion on the sameness of the reasoning 

processes asserted and vindicated . . .60 

Account of logic in the Encycloppcdia Britannica . 63 
Logic another term for reasoning . . .65 

Whately's logic considered . . . . .68 

His analysis of arguments . _ . . . .73 

Nature of syllogism . . . . . .74 

Comparison of' logical and mathematical reasoning . 80 



SECTION III. 

The connexion between language and reasoning in 
general ........ 92 

Approach to mathematical exactness in metaphysical 
sciences, how attainable . . . . .95 

Distinction between mathematics which commence, 
and inquiries which end, with definitions . .96 

Demonstration not always necessary . . . 1 00 

On demonstrative reasoning in physical science . 102 



CONTENTS. 

Advantages of physical science. . 
Works of reference .... 
Cuvier on the stud}^ of natural history . 
Metaphysical science .... 
Metaphysical discussions concern the mean 

words ...... 

Stewart's and Mackintosh's dissertations 
Stewart's estimate of Locke 
Mill's Fragment on Mackintosh . 
Mackintosh on Hartley 
Laplace's Essay on Prohabilities . 
Pla}-fair on Laplace .... 

Laplace on association 

Character of Mr. Austin's work, " The Prov: 

Jurisprudence determined" 
Austin's remarks on demonstration connecte- 

ethics ...... 



ng of 



nee of 
with 



XI 

Page. 

104 
107 

108 
109 

111 
112 
113 
118 
121 
124 
126 
127 

130 

131 



REMARKS. 



It is more than a century since Locke con- 
ceived and maintained, after Hobbes and 
perhaps others, that demonstrative reasoning 
was apphcable to other subjects besides the 
mathematics, and particularly to morahty. 
Doctors Law and Hartley, the disciples and 
successors of Locke, entered fully into his 
views ; and Dr. Hartley especially was fond of 
exhibiting his reasoning in a mathematical 
form, and in some instances has very happily 
applied algebraic formulae to illustrate, I do 
not say to confirm, his trains of moral spe- 
culation. 

Mr. Whewell's Thoughts on Mathematics, 
in which he affirms that mathematics afford 
the best example of practical logic, and the 
elaborate article in the Edinburgh Review, 



"A INTRODUCTORY REMARKS. 

No. 126, on Mathematical Studies, which 
treats generally of the influence of mathe- 
matics upon the intellectual character and 
powers, have in some degree recalled atten- 
tion to the subject, and induced me to offer 
some thoughts upon it, which I trust will not 
appear altogether unworthy of perusal. 

I do this with earnestness, and even anxiety; 
not because I conceive that anything original 
or remarkable will be found in the following 
observations ; for I bear in mind an aphorism 
of Dr. Johnson, " He who tries to say that 
which has never been said before him, wdD 
probably say that which will never be re- 
peated after him ;" but because clear and 
just views on this subject have a close and 
important bearing upon the pursuit of science 
of all kinds, whether physical or metaphy- 
sical ; upon the attainment and diffusion of 
truth ; upon the mental and moral improve- 
ment, and consequently the harmony and 
happiness, of man. These clear and just 
views appear to be absent from the minds 
and writings of many w^hose names are of 
no small account in the Uterarv and scientific 



INTRODUCTORY REMARKS. d 

world, although within easy reach of tiie in- 
quiring, if they will use the glass supplied 
by the plain and manly writers of the true 
English school of philosophy. 

I venture upon it further, because Mr. 
Whewell, in the second edition of his pam- 
phlet, declines going more at length into the 
matters touched upon by the reviewer. He 
has therefore left the field open to any one 
who may dare to enter the lists against that 
formidable and heavily-armed knight. 

The interest wdiich may have been felt in 
the papers alluded to has perhaps already 
subsided, but the subject to which they re- 
late is of permanent importance. Mr. 
Whewell discussed the relative value of 
different modes of pursuing mathematical 
studies, assuming their usefulness and im- 
portance. The reviewer, however, went 
into a much wider field, namely, the 
influence of mathematical studies upon the 
mental powers and character in general. 
And it was his strain of remark, so far as it 
was of a metaphysical character, his obser- 
vations about " two logics," " dissimilar de- 
B 2 



4 INTRODUCTORY REMARKS. 

velopments of thought," and "higher and 
lower faculties," which induced me to review 
the reviewer, and put together a few thoughts 
upon the nature of reasoning and evidence in 
general. With many of the positions of the 
reviewer and of those Vv^hom he quotes, about 
mathematical evidence and mathematical pro- 
cesses, the matter of this Essay will be found 
substantially to coincide ; but if there be 
any truth in his view of the disqualification 
of mere mathematicians for inquiries into 
mental and moral philosoplw, I have en- 
deavoured to approach more closely to the 
sources of that disqualification, or rather so to 
point out the distinction between the nature 
of our thoughts and lang-uage on mathe- 
matical and other subjects, as to furnish 
some useful guidance to reasoners and in- 
quirers in morals and rehgion. 

Moreover, while it seemed to me that the 
reviewer had merely taken occasion from 
Mr. Whewell's pamphlet to propound cer- 
tain semi-German notions, and heap together 
certain Kantian phrases, than which none 
are more utterly distasteful to a healthful 



DEFINITIONS. 5 

English palate, it is perceptible, if I mis- 
take not, that Mr. Whewell himself is not 
altogether so clear and satisfactory as lie 
might be, his own mind being apparently 
tainted in some slight degree with German 
phraseology and metaphysics. But I shall 
not trespass further upon ground which lies 
beyond the limits which it is my present in- 
tention to occupy, nor detain the reader from 
the real subject before us. 

Now in order to perceive the possible ap- 
plication of mathematical or demonstrative 
reasoning to metaphysical subjects, the first 
requisite is to understand exactly the nature 
of that reasoning. In order to do this I ob- 
serve, then, — ■ 

First, That it is the important character- 
istic of mathematical reasoning to proceed 
from definitions. It sets out from these as 
precise data, to which appeal is made in 
every step of the demonstrative process. If 
these definitions be not clearly understood, 
if they be not fully granted and well laid up 
in the mind, in vain does the student or 
pupil attempt to proceed. The foundations 



DEFINITIONS'. 



of the science v/ill then be broken up, the- 
field of mathematical reasoning will then be 
closed, for these are the gate of entrance. 

It affects not the truth of the above 
position to inquire and settle whether the 
definitions are hypotheses or facts ; whether 
they be explanations of terms, abstractions 
of the mind, inductions from observation, or 
assumptions which have no foundation in 
the nature of things ; nor to inquire whether 
the definitions of any particular treatise or 
mathematician, from Euclid and his nume- 
rous editors downwards to Newton and his 
successors, are in every respect the best 
possible, such as suit best the subsequent 
course of reasoning, are most easily ad- 
mitted by the student, or bring most clearly 
before the mind the principle necessary for 
future guidance ; nor to inquire how far the 
postulates and axioms partake of the nature 
of definitions, may be resolved into them, or 
mRj be dispensed with altogether, without 
injury to the study. These may be proper 
subjects for the metaphysician or logician. 
They may, to a certain degree, call for the 



DEFINITIONS. 7 

early attention of the mathematical student ; 
but the tutor who should begin with telling 
his pupil all that has been said or might be 
said about the definitions, postulates, and 
axioms, would probably never get him over 
the asses' bridge. Suffice it, as a matter of 
fact, that when you open any elementary 
treatise on mathematics, Euclid, the conic 
sections, plane and spherical trigonometry, 
and even books of arithmetic and algebra, 
the first objects of careful attention to the 
student are definitions. These are the foun 
dations of his science, the elements of his 
reasoning. To these he must adhere ; and 
if there be anything inconsistent with them, 
confusion ensues, demonstration ceases. 

"It is in this last circumstance (I mean 
the peculiarity of reasoning from definitions) 
that the true theory of mathematical reason- 
ing is to be found," says Dugald Stewart in 
his chapter on Mathematical Reasoning, 
chap. ii. sect. 3, of his second volume of the 
Philosophy of the Human Mind, one of the 
best portions of his writings, yet tinctured 
deeply with his peculiar faults. This is the 



8 DEFINITIONS. 

point upon which he rests ; and the writers 
to whom he refers, and with quotations from 
whom he is so fond of nibbhng, will be found 
substantially to agree with him. 

The following passage from Hobbes' Le- 
viathan is also apt to my puqjose. " To 
the priviledge of absurdity, no li^dng crea- 
ture is subject, but man onely. And of 
men, those are of all most subject to it, 
that professe philosophy. For it is most 
true that Cicero saith of them somewhere ; 
that there can be nothing so absurd, but may 
be found in the books of philosophers. And 
the reason is manifest. For there is not one 
of them that begins his ratiocination from 
the definitions, or explications of the names 
they are to use ; which is a method they are 
to use onely in geometry ; whose conclusions 
have THEREBY been made indisputable." — 
Part i. chap. v. 

The importance of the detinitions admits 
of the following familiar illustrations. It 
happened to me to commence the study of 
Suclid with a youth who stumbled at the 
first definition, — " A point is that which hat li 



DEFINITIONS. il 

no parts, or wliich hath no magnitude." 
" Then," said he, " it is a nonentity, — it is 
nothing." He could not or he would not 
admit such an abstraction ; and he began to 
puzzle himself about the infinite divisibility of 
matter, the nature and extension of ultimate 
atoms, the impossibility of finding a given 
place for that which had no parts, and so on. 
He was not content to take this or any other 
definition as a matter of course, and wait to 
see how far the mathematician would be 
consistent with himself in his subsequent 
reasoning. He was determined to weigh and 
settle the justness of every definition in his 
own mind before he would proceed further ; 
in short, he would concede nothing and dis- 
pute everything. Consequently he never 
took kindly to mathematical studies ; and 
perhaps to this hour he looks upon mathe- 
matics as a multitude of words about non- 
entities, or things Avhich have no real ex- 
istence, and consequently no practical value. 
I by no means imply that in mathematics 
the student is to begin with submitting to 
authority, and not to think about the 
B 5 



iO DEFINITIONS DECIDE 

meaning of the language he uses ; nor 
that the tutor should not be prepared to 
defend his own preliminary statements. Of 
this, perhaps, more hereafter. It is sufficient 
to add at present, that Mr. De Morgan, in 
his work on mathematical studies, published 
by the Society for the Diffusion of Useful 
Knowledge, ranks definition first among the 
characteristics, and, I may say, as at the 
foundation, of mathematical reasoning. 

Secondly, It may appear to many super- 
fluous, but it is important to observe that 
the definitions on which mathematical rea- 
soning depends are definitions properly so 
called ; that is, they are explanations of terms 
— determinations of that sense in which the 
words employed as signs and instruments of 
thought are to be taken, used, and under- 
stood. 

Every one who reads over the definitions 
of Euclid must, I should think, immediately 
assent to this. One definition may be 
better than another of a line, or a straight 
line, of a circle, or of parallel lines ; but 
its superiority can only consist in fixing 



THE MEANING OF TERMS. 11 

more clearly that sense of the word about 
to be used, or that quality in the mind's con- 
ception of the thing signified, (more simply, 
the signification,) which alone is to be pre- 
sent to the mind in its subsequent appli- 
cation of the term. 

Tlie definitions of geometry concern, it is 
obvious, the meaning of the terms point, 
line, straight fine, superficies, angle, triangle, 
circle, and so on, so far as they are de- 
finable. And the student would have 
little occasion to pore over these definitions 
if all the terms were previously familiar 
to him, and all had that fixed and clear 
meaning in his mind ; that is, were the signs 
of those certain ideas of figure for which 
they stand in the mind of the geometrical 
reasoner. 

If Mr. Dugald Stewart had kept Euclid 
open before him when speaking of the 
definitions, I can scarcely imagine he would 
ever have called them hypotheses ; and if he 
had not called them hypotheses, he w^ould 
not have maintained that it is the peculiarity 
of mathematical reasoning to employ hypo- 



12 DEFINITIONS DECIDE 

theses instead of facts as the data on which 
we proceed. See Stewart's Philosophy, 
A^ol. ii. pp. 158, 160. 

Take for example the ninth and eleventh 
definitions. " A plane rectilineal angle is 
the inclination of tw^o straight lines to one 
another, which meet together but are not in 
the same straight line." "■ An obtuse angle 
is that which is greater than a right angle." 
With what propriety can it be said that 
these are h}^otheses? An hypothesis is 
that mode of accounting for certain appear- 
ances which, although probable, remains to 
be verified by future experiments or obser- 
vations ; or it is the supposed cause of certain 
effects, whose adequacy or invariable an- 
tecedency is assumed until disproved by 
further investigation. This is the sense in 
which the term hypothesis is generally em- 
ployed. In this sense the Ptolemaic and 
Tycho-Brahic systems of astronomy were 
hypotheses ; in this sense the theory of 
gravitation, as it first occurred to the mind 
of Sir Isaac Newton, as accounting for the 
phases and motions of the heavenly bodies. 



THE MEANING OF TERMS. 13 

was an hypothesis ; in this sense the midu- 
latory theory of hght is an hjq^othesis. 

Now there is no analogy between this 
meaning of the term hypothesis and the 
definitions of geometry, or of any branch of 
mathematical science. These definitions are 
not imaginary explanations of given pheno- 
mena, nor supposed causes of given effects ; 
they are, as above said, simple explanations 
of terms, or attempts, by the substitution of 
other words in place of one general term, to 
place before the mind, often assisted by 
diagram or sensible representation to the 
eye, that object of thought to which the 
said term is invariably and solely to be ap- 
plied. " Every general term," says Aristotle, 
" is the abridgement of a definition." 
. I observe that Mr. De Morgan, on the 
Study of Mathematics, p. 70, places reason- 
ing from hypothesis second among the 
characteristics of geometrical reasoning. "•' In 
the statement of every proposition," he says, 
" certain connexions are supposed to exist, 
from which it is asserted that certain con- 
sequences will follow. Thus, in an isosceles 



14 DEFINITIONS DECIDE 

triangle, the angles at the base are equal, or, 
if a triangle be isosceles, the angles at the 
base will be equal. Here the hypothesis or 
supposition is, that the triangle has two 
equal sides ; the consequence asserted is, that 
the angles at the base, or third side, v.ill be 
equal." 

Let us remark, however, that still the 
hypothesis implies a clear understanding of 
the words employed, as in the above in- 
stance, isosceles and triangle, both of which 
have been clearly defined and are well under- 
stood. Hypothesis here is strictly and merely 
supposition ; a certain figure or relation of 
lines is supposed or granted to exist, from 
which certain consequences are deduced, 
Tlie reasoning would not be vahd, or there 
would be no reasoning at all, if the terms 
employed did not in the first instance ex- 
actly express the thing intended, — the object 
of thought. Much of common reasoning is 
reasoning from hypothesis in this sense ; 
that is, it consists in supposing certain re- 
lations to exist, and in showing that certain 
consequences follow. 



THE MEANING OF TERMS. 15 

It was not in this sense that Mr. Dugald 
Stewart maintained that " in mathematics 
we employ hypotheses instead of facts" as a 
general proposition ; but rather with intent 
to show that the whole of mathematics 
rested upon assumptions, and therefore dif- 
fered from reasonings which turn upon ob- 
servation, and what he calls facts. He often 
implies that mathematical reasoning remains 
good, though there be no such things in 
reality as points, lines, triangles, circles, and 
squares in the mathematical sense. But if 
it be so, even admitting all this, still let us 
remember that the hypotheses or assump- 
tions, so far as the definitions are included 
in them, are of a certain kind ; namely, that 
certain words shall invariably be associe^ted 
with certain meanings or ideas, and no 
other : for example, that you shall not 
reason about a triangle as if it could possibly 
mean a circle, nor about parallel lines as if 
they were not equidistant at all points. 

" It is not on the definition but the con- 
ception," Mr. Whewell asserts, " that the 
properties and demonstrations are built." 



16 DEFINITIONS DECIDE 

But why separate definition and concep- 
tion ? Are they not virtually the same thing ? 
unless by definition we are to understand 
mere words without signification, little black 
marks upon a piece of white paper. The 
definition is of value solely in fixing, and, as 
it were, embodying the conception. Human 
beings reason with words, which are the signs 
or channels of ideas. You can only convey 
your conception of a straight line or triangle 
to another mind by a definition or descrip- 
tion. It is the object of the definition to single 
out that quality or property in the mind's 
conception of the thing which distinguishes 
it most completely from every other object 
or thing whatsoever ; and which, by being so 
distinguished, and having such settled pro- 
perty, becomes the subject of reasoning. If 
it belonged, for instance, to anything else 
besides an angle to be composed of two 
straight lines meeting together, but not in the 
same straight line ; in these words we should 
have no sufficient definition of an angle. 

In the Appendix to his work on the Con- 
nexion of Number and Magnitude, Mr. 



THE MEANING OF TERMS. 17 

De Morgan makes some admirable and use- 
ful observations on the definitions, postulates, 
and axioms of Euclid ; and thus expresses 
-himself: — "Some of the definitions contain 
assumptions of certain conceptions existing, 
to which names are to be given ; namely, 
those of a point, a line, the extremities of a 
line, a straight line, a surface, the extremi- 
ties of a surface, a plane surface, a plane 
angle, a plane rectihneal angle ; others 
assume the possibility of certain relations 
existing, as will appear from the form in. 
which they are put." 

He afterwards speaks of these as " in- 
definable notions," and places the common 
definitions of them in the light of postulates ; 
thus, " Let it be granted that a point has no 
parts or magnitude, and that we are con- 
cerned with no other property of it, if there 
be any." Again, he speaks of some of the 
definitions, those from the eleventh to the 
fourteenth, and from the nineteenth to the 
twenty-third, as purely nominal, and there- 
fore needing no remark. From the tenor of 
his language, which the reader who is not 



18 DEFINITIONS DECIDE 

acquainted with it will do well to consult, it 
would seem that he considered the geometry 
of Euclid as resting very much on common 
notions (xoivrj swoia) which scarcely admit of 
definition. Nevertheless I do not think that 
his language countenances any separation 
between the conception and what is usually 
considered the definition, but the contrary. 
The object of all the definitions clearly is 
to associate a certain term with a particular 
notion or conception, and thus to fix and 
limit the meaning of the term. In his paper 
on the study of mathematics, Mr. De Mor- 
gan says, (p. 69.) " This {i.e. definition) is 
merely substituting, instead of a description, 
the name which it has been agreed to give to 
whatever bears that description." 

In regard to nominal definition, it is to my 
purpose to quote here wdiat Dr. Whateley 
says in his Elements of Logic, p. 155, fifth 
edition ; and although I begin the quotation 
in the middle of a sentence, no alteration is 
made in its force or meaning : — " all that is 
requisite for the purposes of reasoning 
(which is the proper province of logic) is. 



THE MEANING OF TERMS. 19 

that a term shall not be used in diiferent 
senses ; a real definition of anything belongs 
to the science or system which is employed 
about that thing. It is to be noted, that in 
mathematics (and indeed in all strict sciences) 
the nominal and the real definition exactly 
coincide ; the meaning of the word and the 
nature of the thing being exactly the same. 
This holds good also with respect to logical 
terms, most legal, and many ethical terms." 

Upon the whole we conclude that the 
definitions of geometry settle the meaning of 
terms.* 

Thirdly, These terms are the signs of our 
ideas of figure and quantity, including in the 
latter term number and magnitude (both the 

* Pascal, in his Reflexions sur la Geometrie en General, 
justly observes, however, that many notions are assumed, 
and terms are used in mathematics which are not defined. 
" Cette judicieuse science est bien eloignee de definir ces 
mots primitifs, espace, temps, mouvement, 4galit^, ma- 
jorite, dhninution, tout, et les autres que le monde entend 
de soi-meme." And this must be the case in all reason- 
ing ; for, as definition is merely explaining one term by 
many, it is obvious we might go on defining without 
end, and not advance a step towards any valuable con^ 
elusion. 



20 MATHEMATICAL CONCEPTIONS 

how many and how great, quantus) ; which 
ideas or notions come to us, so to speak, 
originally from without ; i. e. they originate 
in sensible impressions. They are not signifi- 
cant merely of what passes within, or of 
mental states, like the terms memory, the 
will, judgement, attention, and desire, unless 
indeed every sensation, such as of whiteness 
or blackness, be considered a mental state, 
and every idea an affection of the mind. 

Here, perhaps, I am treading upon the 
most doubtful, because metaphysical, ground. 
Right or wrong, however, in what may be said 
under this head, it will not invaUdate what 
has been said about definition and its object. 
It appears to me that mathematical reasoning 
consists in tracing the relations of our ideas 
of figure and quantity by means of exactly 
defined symbols, whether words, diagrams, 
or other symbols, one with another, in re- 
spect of agreement or disagreement, equa- 
lity, or inequality ; and these terms and ideas 
receive clearness and strength by constant 
application and reference to external things, 
or sensible impressions ; and also by their 



ORIGINATE IN SENSIBLE IMPRESSIONS. 21 

observed, clear, uniform, and well-defined 
relation to each other. 

The subject matter of mathematical reason- 
ing may tlierefore be considered to be real 
existencies, with as much justice as the sub- 
ject matter of any other reasoning. For in 
all reasoning, what has the mind before it 
but its own abstractions or notions, and 
terms affixed to those notions ? And who 
can say that circles, angles, squares, lines, 
have not as much foundation in, and refer- 
ence to, things as they exist, as white, blue, 
black, soft, hard, or other quahties of body, 
solid, Hquid, brittle, or elastic ; or the ab- 
stract ideas of space, time, beauty, honour, 
virtue, and so on? Our ideas of number 
and figure are ideas constantly forced upon 
us by sensible objects, and all that fills this 
visible diurnal sphere ; the terms significant 
of these ideas are in constant use and appli- 
cation in ordinary life. They are employed 
by the humblest in station and education 
with uniformity of meaning, with clearness 
and accuracy for their purposes. It is true 
thev mav not know anvthins: of the rela- 



22 DIAGRAMS. 

tioiis and properties of triangles, squares, 
circles, parallelograms, as traced by the ma- 
thematician ; but the mathematician's skill 
and wisdom consist only in having traced 
and studied these relations by means of his 
exact definitions, and by his deeper or more 
frequent meditation on their several con- 
nexions and consequences. The ideas or 
notions of number and figure are common to 
all minds. Attention and instruction only 
are necessary to furnish them with the exact 
definitions and new combinations. In num- 
ber, it is obvious that the terms or figures are 
themselves definitions, or their equivalents. 

It is because the subject matter of mathe- 
m^atical reasoning consists, in our ideas of 
figures and magnitudes or quantities, that 
the reasoning may be carried on by other 
signs than words, viz. sensible diagrams. 
The Arabic numerals, and the notations of 
algebra, are artificial contrivances or abbre- 
viated symbols for tracing the relations of 
quantity as they are wanted, or as those re- 
lations follow from the nature of the con- 
trivances themselves. These diagrams, these 



DIAGRAMS. 23 

figures and notations, are the signs and in- 
struments of the mathematician's or alge- 
braist's thoughts ; and it is because they are 
alwaj^s of a clear and certain nature, and 
bear a uniform, fixed, and definite relation 
one to another, that the geometrical reason- 
ing, and the arithmetical and algebraic pro- 
cesses are the same to every mind. 

Upon this circumstance, namely, the power 
of fixing the attention and carrying on the 
reasoning by means or help of sensible dia- 
grams, Locke fastens, as, of the first import- 
ance, and the great peculiarity in mathema- 
tical studies. 

" That which has given the advantage to 
the ideas of quantity, and made them thought 
more capable of certainty and demonstration, 
is, first, that they can be set down and 
represented by visible marks, which have a 
greater and nearer correspondence with them 
than any words or sounds whatsoever. Dia- 
grams drawn on paper are copies of the ideas 
in the mind, and not liable to the uncer- 
tainty that words carry in their signification. 
An angle, circle, or square, drawn in lines, 



24 DIAGRAMS. 

lies open to the view, and cannot be mistaken : 
it remains unchangeable, and may at leisure 
be considered and exa.mined, and the demon- 
stration be revised, and all the parts of it 
may be gone over more than once without 
any danger of the least change in the ideas. 
This cannot be thus done in moral ideas ; we 
have no sensible marks that resemble them, 
whereby we can set them down." — (Book IV., 
chap, iii., § 19.) 

It matters not that he is the best mathe- 
matician, or arithmetician, who needs least 
the sensible diagram, or the figure on the 
paper ; nor to say, with Mr. Stewart, that 
the figure on paper cannot pretend to that 
precise exactness which is the object of our 
reasoning ; that the line we draw will have 
some breadth, and the circle, however steady 
the instrument and the hand, may deviate in 
some point from equidistance. The most 
skilful reasoners can only have a certain 
idea of visible figure, and of the relation 
of its several parts present to their minds, 
which the less skilful require for facility and 
permanency of reference on the paper. The 



DIAGRAMS. 25 

diagram approaches sufficiently to sensible 
exactness to keep before the mind that qua- 
hty of the figure which is the sole object of the 
reasoning ; and it is sufficient that the more 
nearly the specific figure before us approaches 
to exactness, the more applicable will the 
reasoning be to that figure, — or, more cor- 
rectly, it is only in so far as the figure fairly 
represents the mind's view of its qualities 
that the reasoning applies to it at all. There 
is no such mystery in the most obscure of 
the definitions as to niake us deny their re- 
ference to a certain specific quality of ob- 
jects, that is, to real existencies, in the 
only practical sense of the words. The 
constant application of mathematical rea- 
sonings to the various branches of natural 
philosophy, and the common use of mathe- 
matical terms in the mathematical sense, 
prove the contrary. We speak, for in- 
stance, of the line between one shade of 
colour and another, and length without 
breadth is the only object of the mind's 
contemplation in so speaking. Points and 
angles are words oi perpetual occurrence : the 
c 



26 MATHEMATICAL NOTIONS 

former in the sense of the commencement or 
termination of Unes, without being any de- 
cided parts or given portions of the hne ; 
and the latter in the sense of the meeting of 
two or more hnes together, converging or 
diverging v^dth more or less of rapidity or 
extension. 

It is perhaps of little consequence to de- 
termine whence we get the notions or con- 
ceptions upon which mathematical reasoning 
turns, whilst it is certain we have the notions 
and defined terms appropriate to them, ex- 
cept in so far as it appears that in numerical 
calculations, and in the geometry of Euclid, 
there is a certain verification of the reasoning 
by an appeal to the evidence of the senses. 
In fact, it is hard to divine Vv^hence we get 
notions of figure or quantity if it be not 
from the sight and the touch, or from expe- 
rience, — a word of extensive signification, 
comprehending all the results of observation 
and reflection. Those who say we do not 
get these notions or conceptions from expe- 
rience, would do well to tell us whence we do 
get them, or produce the mathematician 



DERIVED FROM THE SENSES. 27 

upon whom God has not bestowed the five 
senses with which lie has happily blessed the 
rest of mankind. 

I will here venture a remark upon Mr. 
Whewell's language, in his pamphlet on Ma- 
thematical Studies (second edition, p. 32) : 

" I mentioned it," says Mr. Whewell, " as 
likely to make the study of mathematics 
less beneficial as a mental discipline than it 
might otherwise be if the first principles of 
our knowledge be represented as borrowed 
from experience, in such a manner that the 
whole science becomes empirical only. 

" I will not suppose that any person who 
has paid any attention to mathematics does 
not see clearly the difference between neces- 
sary truths and empirical facts, — between 
the evidence of the properties of a triangle 
and that of the general laws of the structure 
of plants. The peculiar character of mathe- 
matical truth is, that it is necessarily and 
inevitably true ; and one of the most im- 
portant lessons which we learn from our ma- 
thematical studies is a knowledge that there 
c2 



28 MATHEMATICAL NOTIONS 

are such truths, and a famiharity with their 
form and character. 

"' This lesson is not only lost, but read 
backwards, if the student is taught that 
there is no such difference, and that mathe- 
matical truths themselves are learnt by ex- 
perience. I can hardly suppose that any 
mathematician would hold such an opinion 
with regard to geometrical truths, although 
it has been entertained by metaphysicians of 
no inconsiderable acuteness, as Hume. We 
might ask such persons how experience can 
show, not only that a thing is, but that it 
must be ; by what authority he, the mere 
recorder of the actual occurrences of the 
past, pronounces upon all possible cases, 
though as yet to be tried hereafter only, or 
probably never. Or, descending to par- 
ticulars, when it is maintained that it is 
from experience alone that we know that 
two straight hues cannot enclose spa.ce, we 
ask, who ever made the trial, and how ? 
And w^e request to be informed in what way 
he ascertained that the lines with which he 



DERIVED FROM THE SENSES. 29 

made his experiment were accurately straight. 
The fallacy is in this case, I conceive, too 
palpable to require to be dwelt upon," 

A meaning of the word empirical has 
crept into our language lately, in conse- 
quence of the freedom with which some phi- 
losophers treat the king's English, and I fear 
also from the bad translation of some Ger- 
man writings, which it was not wont to 
have, as if it v^ere simply equivalent to ex- 
perimental ; whereby we are threatened with 
the loss of a good word for a very important 
idea, namely, that of quackery, or the ob- 
servance of rules drawn from a narrow ex- 
perience, in neglect or defiance of a large 
and true experience. In eight instances of 
the use of the word by our best old English 
authors, which Johnson gives, it is inva- 
riably associated with this latter meaning. 
No fact, which is a fact, can deserve the 
epithet empirical. 

Mr. Whewell would have done well, if 
we do not get our knowledge of the first 
principles, or, as he better expresses it, 
the fundamental conceptions of mathema- 



30 MATHEMATICAL NOTIONS 

tical science from experience, to inform us 
whence or how we do get them. As he has 
not supphed that information, his reader 
may be apt to pause ; and if he be a friend 
of that wise and cautious old gentleman 
hight Experience, he will not easily allow 
the laugh to be turned against him. Just 
so Mr. Dugald Stewart, in his remarks upon 
demonstrative reasoning, says, "It is by no 
means sufficient to account for the essential 
distinction which every person must perceive 
between the irresistible cogency of a mathe- 
matical demonstration and that of any other 
process of reasoning:" "that, in mathema- 
tics, there is no such thing as an ambiguous 
Vv^ord." But Mr. Stewart does not help his 
reader to account for it in any other way. 
Thus he first plunders him of an all-sufficient 
principle, and then leaves him in the dark ; 
nay, he lays it down as his own principle, 
that it is the peculiarity of mathematics to 
reason from definition, as if keeping to him- 
self what he would not allow to another.* 

* See on this the passage of Du Hamel's, quoted by 
the Edinburgh reviewer, p. 427. 



DERIVED FROM THE SENSES. 31 

It appears to me very reasonable to ask, 
" What but experience can show, not only 
that a thing is, but that it must he ?" — a 
very general and perhaps useless proposition. 
— Experientia clocet ; and let every man be 
careful how he limits the extent and value of 
her lessons. As we did not make our own 
senses, nor the external world, we are 
supplied by the constitution of our frame 
with certain conceptions which are natural 
to, and inseparable from, that frame. We 
make words or signs for our conceptions, 
and by use the words become indissolubly 
associated with those conceptions ; and so 
long as we make those signs or words stand 
for those certain conceptions, so long, in 
fact, as being signs they have signification, 
we act the part of rational and consistent 
beings. If a plain man be asked how he 
knows that two straight lines cannot enclose 
space, he may, in his turn, ask the questioner 
whether he ever knew it otherwise ; and so 
may force him to own that constant experi- 
ence taught him that truth ; that nature had 
furnished him with the notions of a line and 



32 EXPERIENCE. 

of straightness, and the words belonging to 
those notions ; that his mathematical studies 
had built upon that experience ; and that, in 
regard to ascertaining that any given lines be- 
fore him were accurately straight, it was clear 
that the straighter they were, in any conceiv- 
able meaning of the term straight, the less 
likely they were to enclose a space. If I 
were asked how I know that in any right- 
angled triangle the square which is described 
upon the side subtending the right angle is 
equal to the squares described upon the sides 
which contain the right angle, my first an- 
swer might be, that I knew it by studying 
the 47th of the first book of Euchd. But the 
study of Euclid forms a small part of my ex- 
perience, which includes all my observations 
and reflections upon the contents of Euchd, 
and all the conceptions gathered from the 
study of the relations traced and traceable be- 
tween the various figures therein the subject 
of meditation. In short, experience, like na- 
ture, is a word of such very comprehensive im- 
port, as containing within itself so completely 
the sources of knowledge and instruction, that 



BEDDOES ON DEMONSTRATION. 33 

whatever does not fall within the boundaries 
of that wide domain, can be nothing short 
of immediate inspiration.* If a friend asks 
me to shoiu him that the thing must be 
so, or in other words to furnish him with 
Euclid's proof, or a mathematical proof of 
the truth of that 47th proposition, that is 
another question ; and then I should recal 
the steps of the demonstration ; and what 
demonstration is, is the matter into which we 
are now inquiring. 

There is a well-known work by Dr. Beddoes, 
on the nature of Demonstrative Evidence, 
which contains many uspful observations on 
the connexion between language and thought, 
in which he endeavours to show that Euclid's 
reasoning begins from experiment and pro- 
ceeds by experiments. There is an awkward- 

* Of course it would be absurd to contend that the 
truths and demonstrations of geometry are lessons of 
mere experience in a sense strictly analogous to that in 
which we apply the term to the observations and details 
of our ordinary daily existence and sensation. We are 
discussing solely the origin of the fundamental concep- 
tions on which mathematical reasoning rests- — -the data 
from which it starts. 

c 5 



34 PLAYFAIR ON BEDD0E9. 

ness in the phrase mental experiments, which 
the Doctor uses, and which might have been 
avoided by a different mode of stating his 
argument or view ; and which seems to be 
this, — that the fundamental notions or con- 
ceptions from which mathematical reasoning 
starts, and to which it appeals, are as much 
the result of experience, and rest as much 
upon the evidence of the senses, and the 
natural meaning of our own words, in con- 
nexion with that evidence, as the funda- 
mentals of any physical science whatsoever ; 
and he instances particularly the axioms, as 
they are called, that " two straight lines 
cannot enclose a space," and " the vvhole is 
greater than its part," 

In a review of a treatise of Leslie's, on 
mathematics, attributed to Professor Play- 
fair, in the twentieth volume of the Edin- 
burgh Review, there are some remarks 
upon this work of Dr. Beddoes, which, 
coming from Professor Playfair, are entitled 
to particular consideration. Playfair sug- 
gests that Beddoes was no great mathe- 
matician. But with submission, this is no 



HARTLEY ON PROPOSITIONS. 35 

answer to Beddoes'" argument, and rather 
too near an approach to the common tac- 
tics of controversial writing, in which the 
reader's attention is diverted from the ques- 
tion, and the pursuit of truth, by some in- 
sinuation against the character or abiUties of 
an adversary. Playfair tells us that " geome- 
trical reasoning is a process purely intellec- 
tual, and resting ultimately on truths which 
the mind intuitively perceives.'" Are we, then, 
to rest here without going further, — without 
venturing to ask what are " truths intuitively 
perceived" ? In what sense this is true, the 
present observations are meant to illustrate, 
and, if I am not very much deceived, will 
sufficiently, or in a great measure, help the 
reader to understand. Meantime I beg to 
call his attention to a remarkable and just 
sentence of Hartley's, in his invaluable and 
profound chapters on " Words, and the Ideas 
associated with them, and on Propositions 
and the Nature of Assent." "Rational 
assent to any proposition may be defined 
a readiness to affirm it to be true, proceeding 
from a close association of the ideas sug- 



36 FORCE OF HABIT 

gested by the proposition, with the idea or 
internal feehng belonging to the word truth, or 
of the terms of the proposition with the word 
truth ;" and then follow some observations 
on geometrical and mathematical reasoning, 
which are as clear, beantiful, and unanswer- 
able, as any observations upon abstract truths 
Avithin the circle of human science and phi- 
losophy. 

But, fourthly, whencesoever w^e get the 
notions or conceptions with which we are 
concerned m mathematical reasoning, I 
think it must be admitted, that habit, i. e. 
the constant recurrence of the same simple 
ideas of number and figure, and the constant 
association of the same terms with the same 
ideas, has much to do with that feehng of 
certainty and satisfaction, that readiness and 
confidence of assent, wiiich we recognise in 
connexion with the processes of arithmetic, 
algebra, and geometry. 

How much there is in habit may be easily 
and irresistibly showm. Thus we say that 
2 and 3 make 5, and the three angles of 
a triangle are equal to two right angles ; 



IN MATHEMATICAL REASONING. 37 

and we feel the truth as we pronounce 
the words. But if we take higher numbers, 
and more advanced propositions, — if we say 
that nine thousand six hundred and seventy- 
three (9673) times seventy-three thousand 
six hundred and nine (73,609) make 
712,019,857, or upwards of seven hundred 
and twelve millions ; or if we take some of 
the propositions relating to proportion in the 
fifth book, or go on to the more abstruse calcu- 
lations in algebra, trigonometry, and fluxions, 
will our assent be so ready? Who will 
assert it ? And why ? — because we are not in 
the habit of attending to high numbers and 
advanced propositions. Doubt, ignorance, 
and difficulty attach themselves to our terms. 
He who has just risen from calculations, or the 
study of mathematics, will feel a confidence in 
terms and propositions which others do not. 
A ready accountant casts up with a glance or 
two a long column of accounts ; he perceives 
the relation of each item to the whole 
amount in a space of time that appears incre- 
dibly short to one wholly unaccustomed to 
such work. Those who are in the habit of 



38 FEWNESS OF TERMS AND PREMISES 

estimating the number of persons in a 
crowded room or assembly, can tell by 
looking at the mass, wdth reference to the 
space occupied, how many may be present 
with much more correctness than another 
who should try for the first time to count the 
heads. So the bare statement of a pro- 
position, and a glance at the diagram, mil 
enable the quick mathem.atician to under- 
stand the whole demonstration, and to re- 
peat the various steps of the process faith- 
fully to another ; while he who is slow at 
combining the ideas of figure, notwithstand- 
ing ever so careful reading of the proof, will 
be still at a loss to perceive its cogency ; and 
will pass from the words to the figure, and 
the figure to the words, without being a 
whit the wiser, or having any distinct idea of 
what he is about, or where he is, present to 
the mind. The elaborate paper of Sir W. 
Hamilton, of Dublin, to the Royal Society*, 
appears a chaos of warring elements, a mere 
jumble of letters and figures to the tyro 

* " On a General Method in Dynamics." — Phil. Trans. 
1834, Pt.ii.,p. 247. 



IN MATHEMATICAL REASONING. 39 

in algebraic studies ; " monstrum horrendum 
inform' ingens cui lumen ademtum;" but to 
that of the learned reader, and to his own 
eye, it appears as the harmonious and beau- 
tiful arrangement of simple* elements, each 
having its due place and force, combining to 
one noble, important, and useful result. 

Further, in geometrical and mathematical 
reasoning the premises are few ; the terms 
employed are few ; and the mind is only en- 
gaged in tracing the relations of a few dis- 
tinct simple ideas, which are fixed by sen- 
sible impressions. The whole vocabulary of 
Euclid may be comprised in a couple of 
pages. Each book turns upon a few defi- 
nitions. The whole volume is filled with 
repetitions of the same terms, with appeals 
to the same brief premises ; attention is 
more or less frequently recalled to each pro- 
position as it passes in review, and which 
ranks, when proved, among the foregone 
premises. The notations of algebra are 
comparatively few ; the letters which stand 
for unknown quantities derive their meaning 
solely from connexion with, and relation to, 
the known quantities, at least in their first 



40 SAMENESS OF TERMS 

use ; and at last from their relation to each 
other, in consequence of an extended mean- 
ing in the symbols, with which meaning, by 
habitual contemplation, the mind becomes 
famihar. Among the fig-ures of arithmetic 
there are but nine units ; after ten you begin 
with new relations of the first nine ; hun- 
dreds are combinations of tens, thousands of 
hundreds, and so on. And with regard to 
the higher numbers, we can always make 
clear their value to the senses ; for though 
we could form not the least notion how many 
men there might be in a field of battle, or 
how many grains of corn in a sack, by 
looking at them in the mass, yet divide 
them into companies of thousands, of hun- 
dreds, and tens, and by this arrangement the 
mind gains a clear and practical sense of the 
number. It is doubtless by understanding the 
number and character, and the due arrange- 
ment of his forces, that a commander-in-chief 
is enabled to dispose of them to the best ad- 
vantage, and form the order of battle. 

Our ideas of number and figure are what 
Locke calls " distinct simple modes ;" and 
however varied in combination or relations, 



AND SIMPLICITY OF IDEAS. 41 

the same signs or terms are invariably con- 
nected with the sam.e conformations of figure, 
and the same relations of number. Put 
down a three-sided figure in lines, or any 
four or more of the Arabic numerals in a 
line, as 4565, and every human being using 
the English language would express the 
relation in the same terms, — w^ould pro- 
nounce the one a triangle, and read the other 
four thousand five hundred and sixty-five. 

" The idea of two is as distinct from that 
of one," says Locke, B. II., chap, xiii., "■ as 
blueness from heat, or either of them from 
any number ; and yet it is made up only of 
that simple idea of an unit repeated ; and 
repetitions of this kind joined together, 
make those distinct simple modes of a dozen, 
a gross, a million." 

Thus also he speaks concerning figure, 
§ 6 : " The mind having a power to repeat 
the idea of any length directly stretched out, 
and join it to another in the same direction, 
which is to double the length of that straight 
line, or else join another with what inclina- 
tion it thinks fit, and so make what sort of 



42 FEWNESS OF TERMS 

angle it pleases ; and being able also to 
shorten any line it imagines by taking from 
it one-half, or one-fourth, or what part it 
pleases, without being able to come to an 
end of any such division, it can make an 
angle of any bigness ; so also the lines that 
are its sides of any length it pleases, which 
joining again to other lines of different 
lengths, and at different angles, till it has 
wholly enclosed any space ; it is evident that 
it can multiply figures, both in their shape 
and capacity, in infinitum ; all which are but 
so many different simple modes of space." 

There does not appear any advantage, but 
the contrary, in the use of the term " mode,'" 
and alternating it with " idea,'" as Locke 
does in this and in other parts of his Essay; 
but whether ideas or modes, it is e^ddent 
they are simple, because they do not admit 
of being resolved into other ideas or notions 
still simpler, but result at once from uni- 
formity in the structure and impressions of 
the senses, which uniformity laj's the foun- 
dation for language and reasoning. 

The simplicity and uniformity of the sen- 



IN MATHEMATICAL REASONING. 43 

sible impressions of space, or figure and 
number, and the comparative fewness of the 
terms or symbols in use in mathematical 
reasoning, constantly associated with the 
same impressions, — terms or symbols w4iich 
are in fact human contrivances for conveying 
those impressions from one mind to another, 
— these things are to be borne in mind, and 
duly weighed, in estimating the nature of 
demonstrative evidence. Nor let any man 
despise mathematical studies, or think them 
a mere ringing of changes upon the same set 
of bells, because the terms employed are 
few, and the original simple ideas few ; 
otherwise, let him despise the EngHsh lan- 
guage, or language in general, because there 
are only twenty-six letters in the alphabet. 
For what endless varieties of thought, — what 
worlds of wisdom, — what vast structures of 
science, are these twenty-six letters, all-suffi- 
cient ! And what would human life be with- 
out them ? 

But we have not yet analysed the nature 
of mathematical reasoning. We have said 
that mathematical reasoning sets out from 



44 FINAL CHARACTERISTIC 

definitions ; that these definitions settle the 
meaning of terms ; that these terms are the 
signs of onr ideas of figure and quantity, of 
numbers and magnitudes ; that these ideas 
are among the simplest, clearest, with which 
our minds and senses are conversant ; that 
the terms in use, and the simple ideas 
to which they are uniformly appropriated, 
are comparatively few ; that the constancy 
of connexion between the terms and ideas, 
that is, habit, has much to do with that 
feeling of assent and conviction to which the 
reasoning gives rise, by which the processes 
are accompanied, as any one must perceive 
who begins to instruct children in arithmetic 
or in geometry. 

But further, fifthly, and lastly, the demon- 
strative quality of mathematical reasoning 
consists essentially in this, — the perception 
of the agreement or disagreement of certain 
ideas and certain terms with other inter- 
mediate ideas and terms, which are used as 
a measure or test of truth, such ideas and 
terms having been previously selected by the 
mind for a measure or test ; in other words, 



OF DEMONSTRATION. 45 

a means or standard of comparison. To 
demonstrate is to show that a certain propo- 
sition not granted to be true is true by virtue 
of some premise previously admitted or as- 
sumed as a criterion of truth, or by virtue of 
some other truth previously demonstrated."* In 
other words, to demonstrate is to discover 
and trace farther and new relations amongst 
our ideas by comparing them one with an- 
other ; which new relations will be of deter- 
minate and constant character, in proportion 
as the intermediate ideas which are used as a 
means of comparison, as a measure, are 
themselves of distinct and constant character 
and value. Or again, it is to show that 
ideas not clearly perceived to harmonize or 
agree, do harmonize, by comparison, with 
other ideas whose agreement is clearly per- 
ceived. Thus in the first proposition of 
Euclid, the sides of a given triangle are 
proved to be equal when they are ail shown 
to belong to the class of lines which radiate 

* The reader may consult the papers on mathematics, 
by Mr. De Morgan, who gives this just account of de- 
monstration. 



46 FINAL CHARACTERISTIC 

from the centre to the circumference of the 
same circle, or of equal circles ; of which 
class of lines equality is previously premised 
in the fifteenth definition. 

This essential quality of demonstrative 
reasoning is thus distinctly laid down by the 
great master Locke, book iv., c. ii. : — 

" The next degree of knowledge is where 
the mind perceives the agreement or dis- 
agreement of any ideas, but not immediately. 
Though wherever the mind perceives the 
agreement or disagreement of any of its ideas 
there be certain knowledge, yet it does not 
always happen that the mind sees that 
agreement or disagreement which there is 
between them, even vvhere it is discoverable ; 
and in that case remains in ignorance, at 
most gets no further than a probable con- 
jecture. The reason why the mind cannot 
always perceive presently the agreement or 
disagreement between two ideas, is because 
those ideas, concerning whose agreement or 
disagreement the inquiry is made, cannot by 
the mind be so put together as to shew it. 
In this case, then, when the mind cannot so 



OF DEMONSTRATION. 47 

bring its ideas together, as by their imme- 
diate comparison, and, as it were, juxta- 
position and apphcation one to another, to 
perceive their agreement or disagreement, it 
is fain, by the intervention of other ideas, 
(one or more as it happens,) to discover the 
a2;reement or disa2;reement Vvhich it searches ; 
and this is what we call reasoning.''' 

Again, § 3 : — 

" Those intervening ideas which serve to 
show the agreement of any two others are 
called proofs ; and where the agreement and 
disagreement is by this means plainly and 
clearly perceived, it is called demonstration." 

Again, § 7 : — 

" In every step reason makes in demon- 
strative knowledge, there is an intuitive 
knowledge of that agreement or disagreement 
it seeks with the next intermediate idea, 
which it uses as a proof ; for if it were not so, 
that yet would need a proof; since without 
the perception of such agreement or dis- 
agreement there is no knowledge. If it be 
perceived by itself, it is intuitive knowledge ; 
if it cannot be perceived by itself, there is 



48 FINAL CHARACTERISTIC 

need of some intervening idea, as a common 
measure to show their agreement or disagree- 
ment. * * * =^ So that to make a thing a 
demonstration, it is necessary to perceive 
the immediate agreement of the intervening 
ideas, whereby the agreement or disagree- 
ment of the two ideas under examination 
(whereof the one is alw^ays the first and the 
other the last in the account) is found." 

It is not without reason that Locke dwells 
upon this ; and he repeats himself in ch, xv. 
of the fourth book on Probabihty, which 
the reader may consult. 

In this analysis of demonstrative or mathe- 
matical reasoning, it is finally to be obseiwed 
that the definitions are used as the primary 
common measures or tests ; they are the 
original ideas or settled notions by means of 
which the relations of other ideas one with 
another are traced, and the agreement or 
disagreement ascertained and settled, and 
by which the new relations, so ascertained, 
become themselves of determinate and con- 
stant character. Each book begins with its 
necessary definitions ; and each proposition, 



OF DEMONSTRATIVE REASONING. 49 

when settled, becomes itself a premise or test, 
by help of which further relations are traced, 
and new agreements or disagreements as- 
certained and fixed. The mind is con- 
tinually reverting to its original simple no- 
tions, builds carefully upon them, and not only 
has a power to retrace, but is very frequently 
employed in carefully retracing every step 
of its progress. Thus we return to the point 
from which we set out, that definition is the 
basis of mathematical reasoning, and gives 
it its peculiarly fixed, clear, and cer- 
tain character. 

The reader who may doubt whether 
this be a correct or perfect analysis of 
mathematical or demonstrative reasoning, 
is requested, by a careful examination of 
mathematical works, to supply the deficiency. 
Let him apply it to the most simple or the 
most abstruse propositions and demonstra- 
tions, and say what essential quality of such 
reasoning has been omitted. 



50 STEWART CONTRASTED 



SECTION 11. 

Having now analysed with all the com- 
pleteness in our power, the nature of de- 
monstrative reasoning, we are prepared for 
the inquiry, wiiether it differs from other 
reasoning, or reasoning in general, in any 
respects or particulars whatsoever. And, 
if it do not so differ, we are then prepared 
for the important inquiry, how the cogency 
and certainty of mathematical science can be 
applied to and obtained in moral, political, 
metaphysical, and religious subjects. 

Now the tendency and almost the object 
of Mr. Dugald Stewart's chapters on mathe- 
matical demonstration, and on the Aristo- 
telian logic, is to draw a broad hne of dis- 
tinction between mathematical reasoning, 
mathematical evidence, and other kinds of 
reasoning, other kinds of evidence. " JNIa- 
thematical definitions," he says, (vol. ii., 
p. 156, 4to,) " are of a nature essentially dif- 
ferent from the definitions employed in any 



WITH WHATELY. 51 

of the other sciences." Agam, p. 157, he 
speaks of " the essential distinction which 
everyperson must perceive between the irre- 
sistible cogency of a mathematical demon- 
stration and that of any other process of 
reasoning." He repeats this idea in various 
places. I need only refer to p. 203, 
where he says, " If the account which has 
been given of the nature of demonstrative 
evidence be admitted, the province over 
which it extends must be limited almost en- 
tirely to the objects of pure mathematics." 

But what says Dr. Whately in his Ele- 
ments of Logic ? 

" One of the chief impediments to the attain- 
ment of a just view of the nature and objects of 
logic^ is the not fully understanding, or not suffi- 
ciently keeping in mind, the sameness of the 
reasoning 2}rocess in all cases. If, as the ordinary 
mode of speaking would seem to indicate, mathe- 
matical reasoning, and theological, and meta- 
physical, and political, &c., were essentially differ- 
ent from each other, i. e. different kinds of reason- 
ing, it would follow that, supposing there could 
be at all any such science as we have described 
logic, there must be so many different species, 
or at least different branches of logic. And such 
D 2 



52 EDINBURGH REVIEWER 

is perhaps the most prevaihng notion/^ — 3rd. ed. 
p. 23. 

Again, p. 25, lie says, — 

" Supposing it to have been perceived that the 
operation of reasoning is in all cases the same, the 
analysis of that operation could not fail to strike 
the mind as an interesting matter of inquiry." 

And thus, p. 50 : — 

" Whatever the subject matter of an argument 
may be, the reasoning itself, considered by itself, 
is in every case the same process; and if the 
writers against logic had kept this in mind, they 
would have been cautious of expressing their con- 
tempt of what they call syllogistic reasoning, 
which is in truth all reasoning." 

Let us contrast with this, — for there is 
nothing more instructive than bringing into 
juxtaposition the different aspects in which 
these recondite matters are presented to our 
attention, — let us contrast with this the 
barely intelligible language of the Edinburgh 
reviewer, p. 413. 

" Now as all matter is either necessary or con- 
tingent, (a distinction Avhich may be here roughly 
assumed to coincide with mathematical or non- 



CONTRASTED WITH WHATELY. 53 

mathematical;,) we have thus^ besides a theoretic or 
general logic, two practical or special logics in 
their highest universality or contrast. 

"theoretical LOGIC. 

Practical logic, as spe- Practical logic, as spe- 
cially applied to neces- cially applied to con- 
sary matters = mathe- tingent matter ::= philo- 
matical reasoning. sophy and general rea- 

soning." 

He says, p. 422,— 

" How opposite are the habitudes of mind which 
the study of the mathematical and. the study of 
the philosophical sciences require and cultivate, 
has attracted the attention of observers from the 
most ancient times. Tlie principle of this con- 
trast lies in their different objects, in their differ- 
ent ends, and in the different modes of considering 
their objects," 

He speaks also of matliematics as " de- 
termining dissimilar developments of thought 
from other sciences," as " not developing the 
higher faculties," as " dependant on the 
lower imagination." 

Again, p. 422 : — 

"Mathematics, departing from certain original 
hypotheses, and these hypotheses exclusively de- 
termining every movement of their procedvire, and 



54 GENERAL INFLUENCE 

the images or vicarious syraljols^ about which 
they are conversant, being clear and simple, the 
deductions of these sciences are apodictic or 
demonstrative; that is, the possibility of the con- 
trary is at every step seen to be excluded in the 
very comprehension of the terms. On the other 
hand, in philosophy, (with the exception of the 
science of logic,) and in our reasonings in general, 
such demonstrative certainty is rarely to be at- 
tained ; probable certainty, i. e. where we are never 
conscious of the impossibility of the contrary, is all 
that can be compassed ; and this also not being 
internally evolved from any fundamental data, 
must be sought for, collected, and applied from 
without." 

" From this general contrast it will be seen how 
an excessive study of the mathematical sciences, 
not only does not prepare, but absolutely incapaci- 
tates the mind for those intellectual energies which 
philosophy and life require. ^Ye are thus dis- 
qualified for observation either internal or external, 
for abstraction and generalization, and for common 
reasoning." 

Now common reasoning we conceive to be 
very bad reasoning ; such reasoning as fails 
to satisfy the man who is seeking for clear 
and exact views, who fears to be misled by 
words, and who remembers that fine phrase- 
ology teaches nothing. It may be observed 



OF MATHEMATICAL STUDIES. 55 

here, that whatever force or justness there is 
in the reviewer's general course of observation, 
it all lies in the word " excessive" — " an ex- 
cessive study of the mathematical sciences." 
And it is perfectly obvious that he who is con- 
versant only with mathematical notions and 
mathematical processes, may be ignorant of 
many other objects of human attention, 
which come nearer home to the business and 
bosoms, the pleasures and pains, of mankind 
at large. He who is always dwelling in 
circles and squares, ellipses and parabolas, 
differentials and integrals, may have a pro- 
portionally confined range of thought. He 
will not understand the feehngs and thoughts 
of other men ; and he may fancy, from the 
habitual association of his ideas, or from his 
determining everything in the same way, that 
he can ascertain the precise quantity of en- 
joyment which a company of aldermen de- 
rive from eating and drinking, by means of 
the differential or integral calculus, and de- 
termine the relative merits of Homer and 
Virgil by the rule and compasses. But what 
then ? Shall matliematical studies not be 



56 GENERAL INFLUENCE 

valued as an essential part of the training of the 
youthful mind ? Is Mr. Whewell's sentiment 
invalidated, that they are the best practical ex- 
emplification and exercise of logic ? If there 
be one mode of stud3dng mathematics better 
than another, shall not a mathematical pro- 
fessor discuss this question, and endeavour 
to settle which is best ? How many sciences 
are there which require for their pursuit, 
comprehension, and enjoyment, a thorough 
knowledge of the higher branches of mathe- 
matics, such as astronomy, optics, dynamics, 
and all those which go under the name of the 
mixed sciences. Who would undervalue 
the hisrhest mathematical attainment when 

o 

applied to these branches of science ; and 
not rather regret, when he sees the mathe- 
matician soaring in the clouds and lost in 
the dim distance of algebraic formulae, his 
inability to follow ? ' ' Non omnes possumus 
omnia." But we can all enjoy and apply 
those practical and simple conclusions, for 
the establishment of which the most pro- 
found mathematical investigations are oft- 
times necessary. 



OF MATHEMATICAL STUDIES. 57 

If the question be, What degree of time 
and attention should be given up to mathe- 
matical studies in a thoroughly comprehen- 
sive course of academic education ? or, How 
far exclusive encouragement should be given 
to high mathematical attainment in an uni- 
versity? (which the reviewer has in part 
raised and discussed,) this may be settled 
without depreciating the importance and 
value of mathematics for the discipline of 
the youthful mind. You have then to take 
into account the great and general purposes 
of education, the whole constitution of the 
human mind, the condition, and wants of 
society at large, the fitness of an individual 
for the particular station which he is de- 
signed to occupy, and the kind of knowledge 
which his meditated profession may re- 
quire. 

It is curious to contrast the reviewer's 
statement of the injurious influence of 
mathematical science in disqualifying for ob- 
servation, either internal or external, for 
abstraction and generalization, with the in- 
tellectual character of Sir Isaac Newton 
D 5 



58 herschel's character 

drawn by Sir John Herschel in his Treatise 
on the Study of Natural Philosophy, p. 271. 

" His wonderful combination of mathematical 
skill with physical research enabled him to invent, 
at pleasure, new and unheard-of methods of in- 
vestigating the effects of those causes which his 
clear and penetrating mind detected in operation. 
Whatever dej)artment of science he touched, he 
may be said to have formed afresh. Ascending by 
a series of close-compacted inductive arguments 
to the highest axioms of dynamical science, he 
succeeded in applying them to the complete ex- 
planation of all the great astronomical phenomena, 
and many of the minuter and more enigmatical 
ones. In doing this, he had every thing to create ; 
the mathematics of his age proved totally inade- 
quate to grapple with the numerous difficulties 
which were to be overcome. * * * Of the optical 
discoveries of Newton we have already spoken ; 
and if the magnitude of the objects of his astrono- 
mical discoveries excite our admiration of the 
mental powers which could so familiarly grasp 
them, the minuteness of the researches into which 
he there set the first example of entering, is no less 
calculated to produce a corresponding impression. 
Whichever way Ave turn our view, we find our- 
selves compelled to l)ow before his genius, and to 
assign to the name of Newton a place in our 
veneration which belongs to no other in the an- 
nals of science. His era marks the accomplished 



OF SIR ISAAC NEWTON. 59 

maturity of the human reason as ai^plied to such 
objects. Every thing which went before might 
be more properly compared to the first imperfect 
attempts of childhood, or the essays of inexpert 
though promising adolescence. Whatever has 
been since performed, hoAvever great in itself, and 
worthy of so splendid and auspicious a beginning, 
has never, in point of intellectual effort, surpassed 
that astonishing one which produced the Prin- 
cipia." 

I refer to this treatise with a strong feeling 
of interest, because it is evident from the 
observations on nomenclature, and on science 
generally, that Herschel's clear English 
mind duly estimates the importance of 
settled terms with settled meanings ; and 
while he dwells on the necessity of having 
exact and uniform standards of measure- 
ment and value, his reader is set upon the 
inquiry into the nature and purposes of 
measures or tests. He who can perceive the 
importance of a proper use of words in 
physical science, must feel that importance 
also in metaphysical. Without it, in fact, 
we can have nothing worthy of the name of 
science. Sir John Herschel would probably 



60 



LOGIC DEFINED. 



smile at the idea of mathematical science 
disqualifying for generalization and abstrac- 
tion or any useful exercise of mind. 

I have indulged in these references to 
Dugald Stewart, Dr. Whately, the Edin- 
burgh reviewer, and Sir John Herschel, 
with a view to place before the reader in an 
easy manner the different lights in which the 
same objects, or objects closely alUed in 
nature and character, are presented to our 
attention, and the necessity of close and 
cautious investigation. 

Now bearing in mind the foregone analysis 
of geometrical or demonstrative reasoning, 
in order to perceive its connexion with logic, 
it is necessary to understand what logic is. 
Is Dr. Whately right or wrong when he says 
the reasoning process is the same in all cases ? 
If he is right, of course it follows that 
mathematical or geometrical reasoning is 
but one illustration or practical application 
of logic. 

I am unable to attach any other consistent 
meaning to the term logic, than that it is an- 
other word — the Greek word — for reasoning. 



LOGIC DEFINED. 61 

As a science it investigates the principles of 
reasoning, or analyses and determines the 
process of the mind in reasoning ; as an art 
it is the practical application or exemplifica- 
tion of the rules so deduced. On this 
point nothing can be clearer and more satis- 
factory than Dr. Whately's observations in 
his preface and throughout his treatise. 

Yet notwithstanding this clearness, and 
notwithstanding Dr. Whately's correction of 
the error of Watts in considering logic as "the 
right use of reason," "a method of invigo- 
rating and properly directing all the powers 
of the mind," a writer on logic in the edition 
of the Encyclopaedia Britannica now in the 
course of publication, says, " Logic may be 
defined as the science of the laws of thought 
considered as thought. This is the central 
notion towards which the various views of 
the science, from Aristotle downwards, gravi- 
tate ; it is the one definition in which others, 
apparently the most opposite, find their com- 
plement and reconciliation." Then, by way 
of elucidating this definition, the writer 
(whom from his use of the term laws, and the 



62 LOGIC AS TREATED 

epithets, contingent, necessary, universal, di- 
rigible, and so on, I could suspect to be the 
Edinburgh reviewer already alluded to) pro- 
ceeds to tell us, first, that logic is conversant 
about thought ; in the second place, about 
thought considered as thought ; and in the 
third place, it is the science of the laws of 
thought, because it is conversant about the 
universal and necessary in thought. 

These are the remarks of a writer who 
comments on the erroneous definition of 
logic, in an article which the editor of the 
Encyclopaedia has reprinted ; an article 
which tells us that " Logic is the art of 
properly conducting reason in the knowledge 
of things, whether for instructing ourselves 
or others ; or it may be defined the science 
of human thought, inasmuch as it traces the 
progress of knowledge ; and that its business 
is to evolve the laws of human thought, and 
the proper manner of conducting the reason, 
in order to the attainment of truth and 
knowledge." And while the writer comments 
on this article, he further tells us that from 
Aristotle downwards the purity of the science 



BY THE ENCYCLOPAEDIA BRITANNICA. 63 

has been contaminated by foreign infusions. 
He speaks of Dr. Wliately's Elements 
as vague and vacillating in its views, its 
doctrines neither being developed from the 
primary laws of thought, nor combined to- 
gether as the essential parts of one necessary 
whole. In short, being desirous to make 
something more of a subject than has ever 
yet been made of it, and to see further 
into things than any one else has seen, he 
plunges into darkness and a wood of words, 

" hunc tegit omnis 

Lucus, et obscuris claudunt convallibus umbrae," 

or, like many of his brethren, he is so 
blinded by the mists of his own land that he 
cannot enjoy the cheerful^ sun and daylight 
loved by the children of the south ; and 
when he is pleased to consider thought as 
thought, he forgets that no one in his senses 
was likely to mistake it for " cakes and ale." 
Indeed, but for the eminence to which the 
Encyclopaedia Britannica aspires, and is in 
part deservedly raised as an authority in the 
sciences, he might be benevolently left to the 
condition and neglect in which the New 



64 



LOGIC AS TREATED 



Poor-Law leaves those who will not help 
themselves when they can. 

Seriously, however, when we talk of the 
science of the laws of thought, do we know 
what we are talking about ? With confidence, 
I answer No. The vdiole subject of meta- 
physics, the whole state of our knowledge 
and language on the nature, qualities, powers, 
and affections of the human understanding, 
as may be inferred from the article meta- 
physics in this very Encyclopsedia, is such, 
that to talk of the laws of thought, to speak 
of primary laws, which implies secondary, 
and universal and necessary, which implies 
particular and contingent, is to talk of no- 
body knows what. What has logic to do 
with the laws by which thoughts come and 
go in the mind of a child or of a maniac ? for 
I suppose a maniac has thoughts, and, if so, 
is subject to laws of thought. True enough, 
all logic supposes a thinking mind ; but so 
does every other science ; so do carpentry and 
masonry ; and wherever we have thinking 
minds, there we have minds subject no 
doubt to what we are pleased to call laws. 



BY THE ENCYCLOPAEDIA BRITANNICA. 65 

But to set the mind hunting after the gene- 
ral laws of thought, under pretence of study- 
ing logic, is to entrap the student into un- 
looked-for difficulties, to leave Aristotle 
utterly in the lurch, to give us our labour 
for our pains, and to bring us, after a 
fatiguing hunt, like Spenser's good knight, 
only to the cave of despair. If logic be the 
science of the laws of thought, what is the 
province of mental philosophy ? I do not 
question that the one touches closely the 
province of the other, but science used for- 
merly to consist in nicely distinguishing 
rather than confounding the things that dif- 
fer, howsoever minute that difference. At the 
risk of appearing merely to reprint what my 
reader may find elsewhere, but what cannot 
be too strongly impressed upon the mind, 
I must use the words of Dr. Whately, and 
say that "the attempt to comprehend so 
wide a field is no extension of science, but a 
mere verbal generalization, which leads only 
to vague and barren declamation. In every 
pursuit, the more precise and definite oux 



66 NATURE OF REASONING 

object, the more likely we are to attain some 
valuable result." 

Without further discussion, I must assume 
that logic is but another word for reasoning; 
and the object of it as a science is to ascer- 
tain the process of the mind, to which we 
specially give that name. Now we have 
analysed the nature of mathematical reason- 
ing, or, in other words, we have examined 
the process of the mind in that reasoning. 
Can we, then, abstract what is peculiar to 
the mathematics, and talk of reasoning in 
general without regard to any particulars ? 
May we not ask what is meant by reasoning 
as a term standing alone ? Is there one 
determinate process of mind to which the 
term reasoning is peculiarly and alone appro- 
priate ? 

Nature and the senses give me the idea of 
a man and of a horse. I suppose the body 
and legs of a horse joined to the breast and 
head of a man, and call that supposition or 
conception by the name of a centaur. Doing 
this, would you say I reason? No, I only 



EXAMINED. 67 

imagine, and give a name to the object of 
my imagination, which are indeed important 
elements of the reasoning powers. But when 
I say all animals have feeling, no vegetable 
has feeling, therefore no vegetable is an 
animal ; you would say I reason, although 
from the very obviousness of the words, and 
from their arrangement, and the smallness 
of the effort of which we are conscious in 
following that arrangement, the portion of 
reason concerned, if we could divide reason 
into measureable portions, is almost too in- 
significant to be worthy of the name. But 
if this be reasoning, what have we ? 

First of all, words, or audible sounds asso- 
ciated with many sensible impressions or 
objects — animals. 

Secondly, These objects classified, and 
viewed in a common relation or under the 
affirmation — having feelings. 

And thirdly. Other objects, viewed under 
a different relation, having no feeling, there- 
fore excluded from this class, no vegetables 
animals, or vegetables no animals. 

In this who can detect any thing but the 



68 



WHATELY S LOGIC 



results or lessons of human experience or 
registered observation, classified, and clothed 
in appropriate language, — that which is 
affirmed of one being denied of the other 
class, — language being to us the means 
and very element of thought, at least of 
thought conveyed from one mind to an- 
other ? and hence the beauty of the Greek 
word T^oyog, which is at once verbum and 
ratio — the audible sound and mental ap- 
prehension. 

I prefer, however, taking a work of au- 
thority like Dr. Whately's as a guide for the 
course of thought which it appears most 
useful and important to pursue. In ana- 
lysing the operation of the mind in reason- 
ing, Dr. Whately says, " It will be found 
that every conclusion is deduced in reality 
from two other propositions, thence called 
premises." He contends there must be two 
propositions, and says, (section third,) of a 
valid argument, "It is impossible for any 
one who admits both premises to avoid ad- 
mitting the conclusion." Then, after giving 
an example of the true syllogism, he says 



CONSIDERED. , 69 

there is this maxim resulting from it, " that 
whatever is predicated universally of any 
class of things, may be predicated in like 
manner of any thing comprehended in that 
class," — the celebrated principle called the 
dictum de omni et nullo of Aristotle. After 
some observations on the substitution of 
letters and symbols for the terms of the 
syllogism, on apparent arguments, on the 
importance of finding a proper middle 
term, on generalization and abstraction, 
he winds up the analysis with the remark, 
" that it consists in referring the term we 
are speaking of to some class, viz. a mid- 
die term, which term again is referred 
to or excluded from (as the case may be) 
another class, viz. the term which we wish 
to affirm or deny of the subject of the 
conclusion." 

With a very strong sense of the value 
of Dr. Whately's Elements, — of the cor- 
rectness, and usefulness of the principles and 
views therein detailed, — it may be permit- 
ted me to observe, that even that work is, 
in some degree, deficient in the rigid 



70 whately's logic 

propriety of language which the subject 
demanded, and which might easily have been 
given to it. The analytical outline of logic 
can scarcely be regarded as a successful and 
complete elucidation of the science. Dr. 
Whately himself calls it an imperfect and 
irregular sketch. 

For as the analytical outline, and the syn- 
thetical compendium, appear in juxtaposi- 
tion, the reader naturally expects that they 
should answer exactly the one to the other, 
the analysis being the resolution of the whole 
into the parts, or, if the reader like it 
better, the tracing of given effects to the 
causes from which they spring ; — the syn- 
thesis, — the enumeration of the several 
parts which combine to make the whole, or 
the advance from the cause to the varied 
effects or consequences. But this corre- 
spondence is by DO means so clear as it 
might have been, — as it ought to be. For in- 
stance : having in the analysis stated that 
the operation of reasoning is in all the cases 
the same, (p. 25, fifth edition,) and that in 
every instance in which we reason a certain 



CONSIDERED. 71 

process takes place in the mind, which is 
one and the same in all cases, Dr. Whately 
opens the compendium by saying, " There 
are three operations or states of the mind 
which are immediately concerned in argu- 
ment." Again, after having in the analysis 
described the process in reasoning as the 
deduction of a conclusion from two other 
propositions, thence called premises, in the 
compendium he says, " Reasoning is the 
act of proceeding from one judgement to an- 
other, founded upon that one, or the result 
of it." These discrepancies may be more 
apparent than real ; they may be of slight 
consequence : but the careful reader is, to a 
certain degree, distracted. And as the great 
object of the study of logic is to clear and 
to brace the mind, — as it is but the athletics 
and gymnastics of the reasoning faculties, — 
as clearness and strength are entirely depend- 
ant on perfect precision in the use of terms, 
— so the teacher of logic should avoid a ver- 
bal discrepancy as fatal to his science, as the 
man under training should avoid diluents 
and laxatives of every kind. 



72 ANALYSIS AND SYNTHESIS. 

I am aware that some may think I have 
drawn too strictly the parallehsm between 
the analytical and synthetical modes ; but, 
after a careful perusal of Mr. Dugald 
Stewart's remarks upon the use of these 
terms in ancient and modern philosophy, 
showing that authority may be pleaded for 
using them in an exactly opposite and mu- 
tually convertible sense, and that he himself 
is at a loss to give them precise meaning, — - 
sometimes confuting in the notes what he lays 
down in the text ; — after reading also what 
Maclaurin says about these modes, in his ac- 
count of Sir I. Newton's discoveries, — I cannot 
help considering them in a very simple and 
obvious light, as different or opposite modes 
of going over the same or a precisely similar 
path ; according to the simile of Condillac, 
one being up and the other down the hill : 
only instead of saying,' with Condillac, that 
as the two methods are contrary to one an- 
other, if the one be good the other must be 
bad, I rather say both may be good, accord- 
ing to the position and view which we as- 
sume, and the walk which, for the time, we 



ANALYSIS OF ARGUMENTS. / <J 

please to take. Dr. Whately, at any rate, 
cannot object to a rigid parallelism, since, in 
his Introduction, he uses the words as they 
are used in chemistry ; and by S5Titliesis he 
appears to understand the enumeration of ele- 
mentary substances, thence proceeding on- 
wards to simple combinations or more com- 
plex substances ; by analysis only resolving 
these last, namely, the complex substances, 
step by step, into their simple elements. 
He should therefore obviously have begun 
his synthetical compendium with a clear and 
explicit enumeration of those very elements 
into which, in his analysis, he had resolved 
the science or art under review. 

But the first and second sections of the 
third chapter and second book, on Argu- 
ments, contain a sufficient analysis of argu- 
ments, which are "Reasonings expressed in 
words ;" and with such reasonings chiefly, if 
not entirely, is logic concerned. " An 
argument," says Dr. Whately, "is an ex- 
pression in which, from something laid 
down and granted as true, {i. e. the premises,) 

E 



74 SYLLOGISM. 

something else {i. e. the conclusion) beyond 
this must be admitted to be true as following 
necessarily from the other." Again, a syl- 
logism, — (and let us remember that every just 
argument may be reduced into the form of a 
perfect or pure categorical syllogism, which 
is sufficiently obvious to the student of logic,) 
— " a sjdlogism is an argument so expressed 
that the conclusiveness of it is manifest 
from the mere force of the expression." 

I conceive this to be a just and sufficient 
account of syllogistic reasoning, taken in 
connexion with what was before said respect- 
ing terms. By logic, the force of all reasoning, 
or the correctness of syllogism, is shown to 
depend entirely upon the degree of exactness 
and comprehensiveness of meaning in the 
terms employed, or, what is virtually the 
same thing, of distinctness in the things 
signified ; and the chief purpose or use of 
the study is to call the attention of the mind 
to those forms of expression, and to fix those 
forms upon the memory, which are always 
essential to strict, legitimate, and convincing 



SYLLOGISM. /O 

inference. Now the value or correctness of 
a syllogism depends , mainly upon one prin- 
cipal term, called the middle term. 

" Every syllogism," says Dr. Whately, 
"has three, and only three, terms ; viz. the 
middle term, and the two terms (or ex- 
tremes, as they are commonly called,) of 
the conclusion or question. The middle 
term (called, by the older logicians, argu- 
mentum.) is that with which each of them is 
separately compared, in order to judge of 
their agreement or disagreement." Again 
he says, " Every argument consists of two 
parts : that which is proved, and that by 
means of which it is proved." And again, 
" The axioms or canons by which the vali- 
dity of pure syllogisms is to be explained 
are these ; viz. first, if two terms agree with 
one and the same third, they agree with 
each other ; secondly, if one term agrees 
and another disagrees with one and the same 
third, these two disagree with each other." 

It is obvious that here we have but dif- 
ferent views or statements of substantial!}^ 
the same thing, namely, the use of one 
E 2 



76 SYLLOGISM. 

term, called a middle or third, as a means of 
comparison between two other terms; in 
other words, a reference of two terms to one 
and the same object of comparison, used as 
a measure or test of their agreement or dis- 
agreement,^ — in short, the celebrated dictum 
of Aristotle, that whatever may be predi- 
cated universally of a class of objects may be 
predicated of every individual comprehended 
in it ; which is analogous to the axiom, or 
common notion of equality, that things 
which are equal to the same are equal to one 
another, or that the whole is made up of all 
the parts, 

A syllogism, to make a homety simile, is 
a kind of two -pronged fork ; the middle 
term is the handle which unites the prongs, 
and enables us to seize conveniently the ob- 
ject of our thought, and feed our mental 
appetite v/ith food convenient for it. 

E. g. All horned animals are ruminant ; 
the ox is a horned animal ; therefore the ox 
is ruminant. Supposing the habits of the ox, 
in particular, unknown or doubtful, but the 
circumstance of its possessing horns evident, 



SYLLOGISM. *7l 

you infer that that may be predicated of the 
ox which you have previously predicated of 
the class to which it evidently belongs ; and 
although the syllogism is evidently of little 
use in cases of experimental knowledge, 
where the validity both of the premises and 
the conclusion depends upon one and the 
same process of observation, yet is it evi- 
dent that the first premise of the above ex- 
ample can only be true by involving the 
truth of the conclusion. 

Horned animals 

\ 

I i 

ruminate the ox 

Wliatever exhibits marks of design must 
have an intelligent creator ; the universe ex- 
hibits marks of design ; therefore the uni- 
verse must have an intelligent creator. 

Design 

\ ': 

I I 

Creator the universe 

All reasoning is included in the term Logic ; 
mathematics is reasoning upon figure and 
quantity ; therefore mathematics is a branch 
of logic. 



78 



SYLLOGISM. 



Logic 

I 



reasoning mathematics 

The whole mystery of syllogism consists, 
therefore, simply in referring the two things 
whose relation you wish to ascertain, to some 
common class signified by what is called the 
middle term ; or, regarding the conclusion 
as a truth you w^ish to demonstrate, the 
demonstration consists in referring it to 
some test, i. e. to some more general pro- 
position, whose truth, whose use as a test, 
is previously admitted, agreed upon, or 
assumed.* 



* My attention was arrested by the ^^gnette attached 
to the original edition of Hobbes' Le\dathan after writing 
the above, as exhibiting a similar idea of logic to the 
eye. 




The connexion between logic and just classification 



SYLLOGISM. 79 

In examining the nature of syllogistic rea- 
soning, it is now evident that you have, — 

In the first place, something laid down, 
granted as true, or assumed as necessary, for 
the subsequent proceeding of the mind. It 
does not affect the truth of this position to 
determine whether you must always have two 
propositions, as in a regular syllogism, before 
a conclusion can be drawn ; or whether the 
mind, having admitted one judgement or pro- 
position, be led on from that one to an- 
other inevitably following from it. Suffice it, 
that in every argument you must begin with 
something laid down, granted, or assumed ; 
in other words, you must have some datum 
or data, as points from which to start, or 
ground on which to rest. 

Secondly. Having something laid down 
or granted, it is the characteristic of all cor- 



proves that Hobbes is about right, though he meets with 
Mr. Stewart's particular reprobation, in saying that 
" when a man reasoneth, he does nothing else but con- 
ceive a sum total from addition of parcels, or conceive a 
remainder from subtraction of one sum from an- 
other," &c. 



80 COxMPARISON OF LOGICAL 

rect or logical reasoning, that the conclusion 
necessarily follows from the premises. It is 
involved in the premises ; it is a consequence 
inevitably connected with them. An absolute 
necessity is the quality of all sound reasoning. 

How then does logical or common reason- 
ing differ from w^iat we call mathematical or 
demonstrative reasoning ? In both we have 
data from which to start, and conclusions 
inevitably resulting, involved in the meaning 
of the terms, i. e. in our conception of the 
things signified. 

Will any man assert that logical reason- 
ing is not demonstrative in the mathematical 
sense of the term? that is, does not con- 
vey a feeling of certainty to the mind in the 
justness and necessity of the conclusion. For 
my own part, I feel no difference in respect 
of absolute certainty. Having admitted that 
" Yvhatever exhibits marks of design must 
have an intelligent author," and that " the 
universe exhibits marks of design," it appears 
to me as inevitable to admit that "there- 
fore the universe must have an intelligent 
author," as to admit that the three angles of 



AND MATHEMATICAL REASONING. 81 

a triangle are equal to two right angles, hav- 
ing admitted that the sum of the angles 
on the same side of a straight line, at the 
same point, is equal to two right angles. The 
conclusion is involved in the meaning of the 
terms : in the latter case it rests upon the 
notion of equality ; in the former, the notion 
of design. It must be confessed, however, 
that the notion of design is not so imme- 
diately an idea of sensation, to use Locke's 
phraseology, as that of equahty. It is an 
idea drawn rather from reflection, or what 
we are conscious of when the mind turns in- 
ward upon itself. In common language, two 
things are said to be equal vv^hen there are no 
sensible marks of difference between them. 
But an object of nature or of art will indicate 
design to an observer very much in propor- 
tion to the observer's power of appreciating 
the end aimed at, and the means employed ; 
and surely it is impossible to separate the 
idea of design from the perception of means 
and ends. 

It is not, then, in the circumstance of start- 
ing from certain data or given principles ; it 
E 5 



82 COMPARISON OF LOGICAL 

is not in the necessity of the conclusion, as 
resulting inevitably from the data, that logical 
or common reasoning, and mathematical 
reasoning, differ from one another. We must 
seek for that difference elsewhere. When, 
therefore, the Edinburgh reviewer, after 
Kant, talks of necessary and contingent mat- 
ter, as distinguishing two sorts of logic, one 
of which he equals to mathematical reason- 
ing, the other to general reasoning, it is a 
talk without meaning, or at least without 
clear and sufficient meaning. For in all logic, 
as in all mathematics, the conclusion is 
equally necessary, equally contingent; equally 
necessary in the sense of inevitably following 
from, or being involved in, the data ; equally 
contingent upon the comprehension or force 
of the terms ; that is, the degree of clearness 
and exactness in the things signified. 

But, again, if in all reasonings you start 
from some data, and in all reasonings 5^ou 
have necessary conclusions, where lies the 
difference between what we call mathema- 
tical and common reasoning? Obviously 
we must seek it in the nature or difference of 



AND MATHEMATICAL REASONING. 83 

the data : and here, perhaps, we shall come 
to see in what sense it is true, if in any sense 
it be true, that in mathematics you have ne- 
cessary, in logic contingent, matter. In ma- 
thematics we have, as I have shown, defini- 
tions, i. e. exact terms, significant of certain 
clear ideas of figure and quantity ; and we 
are employed in tracing the relations of these 
ideas one with another. In all other reason- 
ing we have also terms, and these terms are, 
or ought to be, signs of ideas. But while, 
in mathematical reasoning, we are concerned 
with ideas of figure and quantity, (to say no- 
thing of forces and motion, with whatever else 
can be viewed and treated mathematically,) in 
logic we have every variety of term and of 
idea. There is no proposition of any kind, no 
number of words which can be put together 
so as to form a proposition, to whatever sub- 
ject it may relate, which may not form part 
of a syllogism. It is owing to this compre- 
hensiveness or vastness of logic, in its prac- 
tical application, that its true nature is so 
little understood. In all reasoning, in all 
thought, as communicated from one mind to 



84 COMPARISON OF LOGICAL 

another, we have, firstly, the terms ; se- 
condly, the things signified : but the sole 
question is, how far the terms are exact, and 
the things signified clear. 

To put this in another light. If, as we 
have endeavoured to show, the essence of 
logic, of all arguments or syllogisms, consist 
in the reference of tw^o terms to one and the 
same common term or object of comparison, 
used as a measure or test of agreement or dis- 
agreement ; and if, as we have also shown, the 
essence of mathematical or geometrical rea- 
soning consist in the perception of the agree- 
ment or disagreement of certain ideas and 
terms, by means of other intermediate ideas 
and terms, previously admitted as a test ; it 
remains only to inquire what are the peculiar 
measures, or what is the excellence of the 
tests, in mathematical reasoning; for by 
these it obtains its character of force and 
clearness, that is, of demonstration. 

Have you in mathematics better data, bet- 
ter measures, than in other subjects ? The 
only answer is, that in mathematics you are 
conversant with ideas of figure and quantity, 



AND MATHEMATICAL REASONING. 85 

and that you have certain well-defined terms 
always associated with the same simple ideas 
or impressions ; the miiformity in the struc- 
ture of the senses, and in the sensible impres- 
sions made upon them, laying the foundation 
for that peculiar constancy of language, and 
consequent clearness of reasoning, which we 
recognize in connexion with figures and 
numbers. 

It follows, therefore, that it is not in any 
theories about generalization and abstraction, 
it is not in any difference between the higher 
faculties and the common faculties, that you 
are to seek for an explanation of the cogency 
of mathematical and the weakness of other 
reasoning. It is because your abstraction is 
so comparatively easy, and your generaliza- 
tion so complete ; and because you have cer- 
tain exact terms and symbols, used as known 
and admitted measures, or criteria of proof, 
about the application of which there neither 
is nor can be a possibility of doubt, that 
mathematical reasoning is so satisfactory.* 

* Perhaps this comparative easiness of the abstraction 
and generahzation in the mathematical sciences is the 



86 COMPARISON OF LOGICAL 

If in that reasoning there were the shghtest 
ambiguity in terms ; if an angle were mis- 
taken for a chord ; if A B were mistaken for 
any other hne than that intended in the 
geometrical proof ; the reasoning would be 
vitiated, the demonstration lost, the link of 
concatenation broken ; just as in arithmetic, 
if 3^on mistook a 3 for a 5, your answer 
would be wrong ; or, in more dignified phrase, 
your postulatum would exhibit an erratum. 
So, in logic, if the sense of your terms be 
changed in the premises and conclusion, the 
force of the syllogism is destroyed ; the bolt 
of the stable is drawn, and the horse gone. 

Think not there is no classification (and 
wherever classification is, there is of course 

main thing intended by the revievv^er when he speaks of 
the lower faculties only being employed and developed in 
■the study ; but the word " faculties" makes sad havoc in 
the writings of some philosophers, who would lead' you 
to suppose man had as many distinct faculties as a cen- 
tipede has legs. A faculty is simply a power; every 
action and every thought may be attributed to a distinct 
faculty for that action and that thought ; and we may, if 
we please, talk of a faculty for sitting and a faculty for 
walking ; but we gain confusion and lose distinctness by 
such phraseology. 



AND MATHEMATICAL REASONING. 0/ 

abstraction and generalization,) in mathema- 
tics ; for in the very first proposition you 
prove your triangle to be equilateral, when 
the sides are shown to belong to the class of 
lines which radiate from the centre to the 
circumference of the same circle, or of equal 
circles, of which class of lines you have pre- 
mised equality as the characteristic ; and he 
who should dispute this premise must look 
out for another proof of the first proposition 
than that with which his Euchd has furnished 
him. 

It is to be observed particularly, that the 
definitions in geometry, and, I think, in all 
the branches of mathematics, are classifica- 
tions or abstractions of a very simple kind : 
in geometry, of certain sensible impressions, 
derived partly from sight and partly from 
touch. Thus, a line is defined, " Length 
without breadth." It has been suggested 
that the words " without breadth" are unne- 
cessary ; but their use is perhaps expedient, 
in order to confine attention to that which is 
exclusively the object of the thought or rea- 
soning. If any "given fine before us have 



88 COMPARISON OF LOGICAL 

sensible breadth, that breadth forms no part 
of or consideration in the reasoning. Now 
more or less of what we call length is found 
in all visible and tangible objects. There is 
not an object which we see or handle that 
has not some outline, straight or curved. To 
this quality of objects, common to all, but 
peculiar to none, we give the name hne or 
length. It is an instance of abstraction, sim- 
ple and complete ; and when w^e have lines 
of a certain kind, we give them a correspond- 
ing denomination, as straight, curved, waving. 
It is not however with lines, in this very ge- 
neral or abstract character, that geometrical 
reasoning has much to do ; but it is concerned 
with lines in more precise and limited cha- 
racter ; as they are related to each other in 
a certain fixed, perceptible, or conceivable 
manner : for instance, as perpendicular or 
inclined, as • meeting in a point and forming 
angles, as parallel or equidistant, as forming 
squares, circles, elhpses, parabolas, or other 
curves, with fixed properties and relations. 
Hence figures are classed as triangles and 
parallelograms, as equilateral and equiangu- 



AND MATHEMATICAL REASONING. 89 

lar, as pentagons, hexagons, polygons, and 
so on. And with regard to all these classes 
of figures, or definite arrangements of lines, 
the mathematical reasoning is strictly syllo- 
gistic ; as in the fifth proposition of the first 
book, the proof of the equality of the angles 
at the base of an isosceles triangle turns upon 
bringing the angles in question within a cer- 
tain class, viz. the class of angles subtended 
by equal bases, in triangles which have two 
sides of the one equal to two sides of the 
other, of which equality is demonstrated in 
the fourth proposition : and let us remember 
that every proposition in Euclid is demon- 
strated as true, not merely of the individual 
diagram before the student, but of its class, 
of which class the said diagram is, in respect 
of the reasoning, a perfect and sufficient ex- 
ample. Thus the angles of all triangles are 
equal to two right angles, whatever be the 
length of the sides, whether they be right- 
angled or obtuse ; whether the lines be black 
or blue ; whether it be the triangle on the 
paper, or a supposed triangle, formed by 
lines conceived to meet in the centres of the 



90 COMPARISON OF LOGICAL 

earth, the sun, and the moon. It is the 
simplicity and perfection of the classes ; the 
accuracy with which every term marks and 
defines the class ; and the never-faihng con- 
nexion between the terms and the sensi- 
ble impressions, and the ease and certainty 
with which the sensible impressions lead to 
and support the mathematical conceptions 
and definitions ; these things help to make, 
if they do not, as I conceive, themselves 
make, the proof so cogent and the assent 
so firm in geometry. 

The dependence of the reasoning upon a 
clear apprehension of the definition, starting 
from it and adhering to it, becomes still 
more clear, if we look at the subject of pro- 
portion upon which Mr. Whewell has made 
some but not very distinct remarks. The 
fifth book of Euclid, which treats of pro- 
portion, Mr. De Morgan calls, in con- 
junction with Aristotle's logic, the most in- 
disputable treatise that ever was written. On 
the other hand, Leslie, in the fourth Preli- 
minary Dissertation to the Encyclopaecha 
Britannica, says that it cannot possibly be 



AND MATHEMATICAL REASONING. 91 

taught. The whole difficulty seems clearly 
to lie in the necessity of enlarging the mind's 
view of proportion, previously and strongly 
associated with numbers, i. e. with arith- 
metical proportion, to magnitudes whose re- 
lation to each other cannot be expressed in 
numbers ; and this difficulty can only be 
overcome by the assiduous study of such 
magnitudes, and of a book, or books, in 
which such magnitudes and their relations 
are brought before the mind. The mind, by 
the study of the subject, grows to the ap- 
prehension of the definition, which is a ge- 
neral principle or view of a certain mutual 
relation of magnitudes, involving the truth of 
the propositions to which it is afterwards ap-^ 
plied. 



92 LANGUAGE AS CONNECTED 



SECTION III. 

Having now shown that the object of the 
science of logic is to call the attention to 
those forms of expression which are essential 
to valid arguments, in which the conclusion 
is necessarily involved in the premises, and 
the mind is led to perceive a connexion or 
relation which it did not before perceive 
between its ideas and terms, — that it resolves 
itself very much into just classification ; 
having also shown that mathematical reason- 
ing owes its clearness and cogency to the 
simplicity and clearness of its subject mat- 
ter, — its abstractions and classifications, or 
relations of figures and quantities, being 
marked by defined terms, which are the 
media of mathematical proof, — we have to in- 
quire how far other notions, besides those of 
figure and quantities, are susceptible of exact 
definition and exact language, and thereby 
of exact comparison one with another. This 
is the sum and substance of the question, — 



WITH GENERAL REASONING. 93 

the susceptibility of exact comparison be- 
tween our notions of other subjects than 
figure and quantity ; subjects less connected 
with sensible impressions, and in which our 
reasoning cannot be assisted or verified by 
an immediate appeal to the evidence of the 
senses. 

I may be as certain, and doubtless I am, 
that there Hved a celebrated orator named 
Cicero, at Rome, as that the angles at the base 
of an isosceles triangle are equal to one an- 
other ; but it is evident that the ideas asso- 
ciated with the words Cicero, celebrity, ora- 
tory, Rome, are of a far more varied and com- 
plex character than the ideas associated with 
the terms of the above or any mathematical 
proposition ; and of a hundred persons who 
w^ill equally readily assent to the historical or 
moral proposition, the ideas associated with its 
terms will differ by a thousand modifications 
and varieties. Here, then, lies the difficulty. 
Forms and magnitudes visible to the eye, 
and weighed by the hand, can be compared, 
and their exact difference can be estimated 
and described. But who shall compare and 



94 LANGUAGE AS CONNECTED 

estimate tlie exact difference between two 
tastes and two sounds ? and how indefinite 
must those terms remain which are not only- 
associated with a number of such sensible 
impressions incapable of comparison, but 
with trains of sensation and emotion, the 
traces of which pass away with the moment 
in wdiich they have birth, and which are, 
perhaps, wholly similar in no two existing 
minds. 

It is often said, by way of distinguishing 
mathematical science from all other kinds 
of reasoning, that the mathematics are hu- 
man contrivances for attaining human ends; 
and that in the reasoning v/e are only evolving 
the nature and properties of these contri- 
vances. Thus in numbers, we have fractions 
and decimals ; in algebra, equations and roots ; 
so in the affairs of life w^e have invented 
common measures with which to compare 
magnitudes and quantities : thus we have 
the foot, the yard, the pound weight, the 
ton, the pint, the bushel ; by means of which 
quantities, distances, and magnitudes, are re- 
lativelv ascertained and settled. So we have 



WITH GENERAL REASONING. 95 

thermometers, barometers, and chronometers, 
for measuring atmospheric weight, heat, and 
time. These are some of the received mea- 
sures or tests for ascertaining and determin- 
ing the degrees of difference by which one 
quantity varies from another, of whatsoever 
can be measured or tested. 

Let us, hov/ever, remember that all lan- 
guage is a human contrivance for expressing 
human thoughts in all their wide relations 
and variety ; and although the language which 
we daily use may have, and most general 
terms have, very different ideas associated 
therewith in different minds, yet reasoners 
have a certain power over its use. We can 
examine, control, and the exact reasoner 
always seeks to control by examination and 
reflection, the associated ideas. It should 
seem, indeed, that where we have no com- 
mon measures or exact standards of compa- 
rison, we cannot come to any exact conclu- 
sions. And this must be the case in all ques- 
tions of degree, in things that cannot be re- 
duced to measure ; where we must be con- 
tent to use our comparatives and superlatives 



96 LANGUAGE AS CONNECTED 

indefinitely, and satisfy ourselves with the 
old maxim, ' De gustibus non disputandmn.' 
But it is in our power to approach to exact- 
ness, (in mathematical phrase, as nearly as lue 
please,) by using well-defined terms associated 
with uniform sensible impressions, and by dis- 
tinguishing things that differ in kind, if not 
in degree. Y\^e may scrupulously avoid un- 
necessary changes of terms, when the subject 
matter of our reasoning is the same ; leave 
nothing to be understood which it is possible 
to express ; and beware of using relatives 
and pronouns, to which there is no clear or 
certain antecedent. 

It is also often said, by way of further dis- 
tinguishing the metaphysical sciences from 
the mathematical, that the latter turns upon 
human abstractions or hypotheses, and you 
begin with definitions ; whilst the former 
turns upon facts, and you end with definitions. 
But is this an adequate distinction ? Every 
one, doubtless, must be aware of a certain 
difference between what are called inquiries 
into facts, and the pursuit of a train of 
mathematical reasoning. But what is that 



WITH GENERAL REASONING. 97 

difference? What is the process of mind 
in the two cases ? In mathematical pro- 
positions you have all the facts before you, 
and are deducing logically the consequences 
which flow from the acknowledged data. 
In what are called inquiries into fact, you 
are generally testing the correctness of some 
verbal statement, which indicates the ex- 
istence of some but perhaps by no means 
clear impressions upon the human mind ; or 
you are seeking to supply the want of impres- 
sions on your awn senses, by considering 
the nature and evidence of the impressions 
on the senses of others. These impressions 
you gather from what is called testimony ; 
and testimony introduces all the ambiguities 
and difficulties of language. Here you must 
have various methods of examination or 
tests of truth dra^vm from experience and 
suited to the particular case under exami- 
nation, into which it is unnecessary to enter 
at length. 

It is often difficult, we say, to ascertain 
what is the fact. In general all the difficulty 
arises from, or is increased by, the indefinite- 

F 



98 LANGUAGE AS CONNECTED 

ness of the terms in which the statement of 
fact is enwrapped by the witnesses, and the 
want of care, caution, and skill in observing 
and registering observations. The extreme 
rapidity with which the human mind min- 
gles its own inferences with its observations 
or sensible impressions ; the confidence 
with which it attaches imaginary or expe- 
rienced causes to the perceived effects, and 
the difficulty of dissociating the one from 
the other, give rise to various embarrass- 
ments not merely with regard to reliance 
upon the statements, but in ascertaining 
even the meaning of others, — embarrass- 
ments which most writers of reputation in 
the abstruse subjects of morals and religion 
rather increase than help to overcome. 
However, " it is possible," says the sweet- 
minded Hartley, " for two persons of intelh- 
gent and candid minds to understand one 
other." 

Then, as to ending with definitions, the 
metaphysician who talks of that as his end, 
while it is the mathematician's beginning, 
should remember that this is only another 



WITH GENERAL REASONING. 99 

mode of stating that his metaphysical dis- 
cussions are disputes about words. As such 
no doubt they have their importance, the 
question being, What are the ideas associated 
with, or the mental phenomena to be classed 
under, certain terms which exercise exten- 
sive sway over the thoughts, feelings, and 
conduct of men? Let the metaphysician, 
then, take care of the road by which he 
proposes to pursue this end. Let him take 
heed whilst he is in pursuit of a proper defi- 
nition of a certain term, of such a definition 
as will be acceded to by the student, that he 
does not embarrass himself and his reader by 
a multitude of other still less defined or 
definable expressions, which thicken the 
darkness and the difficulty, and conduct to 
dismay and to despair. 

These simple observations will, I conceive, 
go some way towards illustrating the ques- 
tion, in what manner, if in any manner, 
demonstrative reasoning may be connected 
with and obtained in metaphysical and 
moral science. Logic calls our attention 
to the nature and force of terms ; to those 
f2 



100 DEMONSTRATION 

forms of expression which are esseritialty 
connected with vaUd reasonings — reason- 
ings to w^iich the mind assents through 
the very nature of the terms. Mathe- 
matical science exhibits the perfection of 
such reasoning and such terms. It works 
with symbols significant of certain clear and 
uniform notions of figures and magnitudes. 
In its higher branches it is indeed the very 
science of symbols ; its results arising out of 
the conceived and admitted nature of the 
symbols themselves, and their uniform rela- 
tions one with another. 

Before saying a few^ words on the applica- 
tion of demonstrative reasoning to physical 
and metaphysical science, I cannot forbear 
remarking how groundless is the common no- 
tion that demonstrative reasoning is stronger 
than all other kinds of evidence, — than the 
evidence of the senses or of testimony ;* a 

* There is some logical impropriety in connecting the 
words demonstration and evidence, and in talking of 
demonstrative evidence. In demonstrative reasoning we 
trace the harmony subsisting among om- agreed princi- 
ples or admitted notions and conceptions, and the con- 
sequences or connected notions. The senses (and testi- 



NOT ALWAYS NECESSARY. 101 

notion which makes some men call for 
demonstration where it is evidently absmxl 
to make that call, unless it be remembered 
that the nature of every proposition must 
determine the nature of the demonstration, 
or the kind and criteria of proof. But who 
shall undertake to demonstrate by any ad- 
mitted criteria of proof that milk is wdiite 
and the sky sometimes appears blue ? These 
are simple instances of names attaching to 
the objects of sense, and all communion of 
mind depends on agreeing to give similar 
names to similar impressions. And what 
folly would it be when my servant testifies 
that a crowd is in the streets, or Mr. Smith 
is in the study, to ask him to demonstrate 
these affirmations and propositions. The 
conduct of life depends upon notions and 
habitual statements which require no demon- 
stration, no laboured process of inquiry and 
proof; and with the general language of 
human intercourse, the associated ideas are 

mony, which must be ultimately referred to impressions 
on the senses) evidence the existence of certain causes of 
sensation external to ourselves, independent of our ovv^n 
minds, and supply the fundamental conceptions from 
which we reason. 



102 PHYSICAL SCIENCE. 

sufficiently clear and uniform for the harmony 
and happiness and the ordinary wants of 
society. But when men aspire to what is 
called science, then must they take heed of 
language, as the ladder on w^hose rails the 
foot must rest in every step of the ascent. 

Now with regard to physical science, it is 
obvious man is simply the observer and 
registrar of external nature. " Homo est 
nature minister," is the short and obvious 
maxim of the natural philosopher, and it 
is evident that the sciences which w^e call 
inductive merely give us the combined re- 
sults of human observation in various spe- 
cified departments, arranged, classified, and 
marked, and mathematical science is essen- 
tial to perfect our observations. 

As we range the walks of time and space, 
every new object or relation which presents 
itself to attention, a new term, significant 
of that object or relation, must be given. 
That same term will suffice for the same 
object or its counterpart, when the mind 
meets with it again. And the uniformity 
and variety in nature gives rise to the uni- 
formity and variety in the structure and use 



PHYSICAL SCIENCE. 103 

of human language, which, as a whole, 
may be considered a mystical radiation from 
nature, imprinting its pictures upon the 
subtle ground-work of the mind. A term 
becomes general by being apphed to many 
similar objects or impressions. The uni- 
formity in the objects and processes leads 
us analogically to apply the term laws to 
the phenomena and processes of nature, as 
science in its various departments reveals 
and registers the order, succession, and 
character of these phenomena. Each sci- 
ence having its own pecuHar phenomena to 
register, requires its own pecuhar terms ; 
nor can we, where hew objects of perception, 
new relations of thought, are to be expressed 
and held forth to the contemplation of 
the mind, object to many new and there- 
fore hard terms. But in these days the 
student of nature must be warned against 
supposing that a mere knowledge of terms 
is scientific knowledge, although it must be 
admitted as an indispensable and an un- 
avoidable part thereof. The learner of a 
new science cannot but be as the child, whose 



]04 PHYSICAL SCIENCE. 

understanding grows to the meaning and 
right use of the language of the world around 
him, nor must he complain of difficulties and 
impediments. Happy the learner, however, 
w^ho is in the hands of judicious guides, wiio 
consider terms subsidiary to instruction in 
things, w4th whom books are, as they 
assuredly ought to be in physical science, 
not the substitutes for the companionship of 
nature, but the aids to interpret her lessons, 
and to observe and to arrange her instruc- 
tions. The botanist must keep in the fields 
and the garden; the chemist in the laboratory; 
the geologist in the quarry, by the hill side 
or under the cliff; the astronomer must sweep 
the heavens with his glass, and report to 
others 

" Of fields of radiance, whose unfading light 
Has travell'd the profound six thousand years. 
Nor yet arrived in sight of mortal things." 

And it is the beauty of physical science, 
wdien legitimately and lovingly pursued, that 
it calls us into communion w^ith the Creator 
as he reveals himself in his w^orks, and away 
from the perverse disputes and vain jang- 



PHYSICAL SCIENCE. 105 

lings of men. In physical science there is 
comparatively little of tiresome useless argu- 
mentation ; the facts on which the classifica- 
tions and conclusions rest are evident to those 
senses of whose use not even the Fall has 
deprived the children of Adam. And though 
the senses may sometimes deceive us from 
a kind of natural difference in keenness or 
constitutional imperfection, or through hasty 
inferences and casual associations, yet the 
philosopher who builds his system of science, 
physical or metaphysical, upon other ground, 
who thinks the root of the tree of life and 
knowledge is not in that plain but all-sup- 
porting soil, had better return at once to the 
speculations of the schoolmen, and puzzle 
himself with inquiries into the necessary 
attributes of spirits that have never inhabited 
a body, or dilate upon that pleasant and 
edifying subject, " cliim.?era bombinans in 
vacuo." 

Vanity, partiality for their own habits of 
study, and their own modes of classifica- 
tion, may even in physical science lead men 
to differ widely and warmly ; but the differ- 
F 5 



106 PHYSICAL SCIENCE. 

ence will hardly be very sore, if unconnected 
with any worldly interests. Questions of fact, 
and questions of the meaning of terms in 
most matters of pure science, would be settled 
without length of debate, if men did not wish 
to appear wiser than they are ; if they were 
content patiently to learn and mildly to in- 
struct ; if they sought the knowledge and 
love of nature rather than the estimation of 
men ; if they looked upon themselves as 
mutual interpreters, and mutual servants of 
the will of Him, who has made of one blood 
all nations of the earth to dwell together 
upon its face, and bound the vast family of 
man together by the strongest ties of mutual 
interest, making it their chief and noblest 
happiness to benefit and assist each other. 

In the pursuit of physical science it would 
be very easy to repeat some of the principal 
rules to be observed ; but they are already in 
various forms and in abundance before the 
studious and inquiring world. To observe and 
register carefully, not to generalize too fast, to 
take special care of your premises before you 
arrive at your conclusions, to build the super- 



WORKS OF REFERENCE. 107 

structure of system upon the solid ground- 
work of clear and well-ascertained facts, and 
with the strong masonry of well-defined and * 
chastised language — the cement of human 
knowledge — these are the short and simple, 
but universal rules for all philosophy. The 
Novum Organum of Bacon ; the first 
of the preliminary dissertations prefixed to 
BufFon's Theorie du Monde ; the admirable 
Reflexions sur la Geometric, in the Pensees 
de Pascal ; Hartley's invaluable Observations 
on Propositions and the Nature of Assent ; 
the last book of Locke's Essay on the Human 
Understanding ; Herschel's Treatise on the 
Study of Natural Philosophy ; Dr. Whately's 
Logic ; some of the remarks of Laplace, 
in his Essay on Probabilities, to which I 
shall hereafter advert, and of Cuvier in the 
Preface and Introduction to his Animal 
Kingdom ; and Mr. De Morgan's treatises 
published by the Society for the Diffusion of 
Useful Knowledge ; — are among the works of 
most importance in connexion with the subject 
of this Essay. The following passage from the 
preface to Cuvier's first edition of his Regne 



108 EXTRACT FROM CUVIER. 

Animal will be acceptable to the reader who 
is not previously acquainted with it, and 
bears closely upon our present course of 
thought, " The habit necessarily acquired 
in the study of natural history, of the mental 
classification of a great number of ideas, is one 
of the advantages of that science which is sel- 
dom observed, and which, when it shall have 
been generally introduced into the s^^stem of 
common education, will become perhaps the 
principal one. By it the student is exercised 
in that part of logic which is termed method, 
just as he is by geometry in that of syllo- 
gism ; because natural history is the science 
which requires the most precise methods, as 
geometry is that which demands the most 
vigorous reasoning. Now this art of method, 
once well acquired, may be applied with in- 
finite advantage to studies the most foreign 
to natural history. Every discussion mi- 
plying a classification of facts, every inquiry 
w^hich demands a distribution of materials, is 
performed according to the same laws ; and 
the young man who had cultivated this 
science merely for amusement, is surprised, 



METAPHYSICAL SCIENCE. 109 

when he makes the experiment, at the facih- 
ties it affords him in disentanghng all kinds 
of affairs. It is not less useful in solitude. 
Sufficiently comprehensive to satisfy the 
most powerful mind, sufficiently various and 
interesting to calm the most agitated soul, 
it consoles the unhappy, it soothes animosi- 
ties. Once elevated to the contemplation of 
that harmony of nature irresistibly regulated 
by Providence, how weak and insignificant 
appear those causes which it has been pleased 
to leave dependant on the arbitrary will of 
man. How astonishing to behold so many 
examples of fine genius consuming them- 
selves so vainly for their own happiness, or 
that of others, in the pursuit of empty specu- 
lations, whose very traces a few years suffice 
to sweep away." 

But is the motto "homo est naturse mi- 
nister" applicable solely to physical and not 
to metaphysical, moral, and political sci- 
ence ? Can we study the human mind and 
human interests without a careful observa- 
tion of the phenomena or facts ? Is man to 
interpret his mental and moral constitution 



110 METAPHYSICAL SCIENCE. 

by means of some a ??norf principles, which 
are to be received, without question, upon 
the ipse dixit of this or that philosopher? 
There is indeed a growing perception of the 
importance of the Baconian method, in moral 
and political science as well as in other de- 
partments of human inquiry, notwithstand- 
ing the tendency of many productions of 
eminence to draw away men's attention from 
the palpable instruction of nature to the 
cloudy, obscure, inconsistent, and unmean- 
ing language of the schools. But a slight 
acquaintance with the most popular works 
on ethical and metaphysical philosophy, will 
lead to the conclusion that they are sadly 
deficient in that exact language which is 
the essential characteristic of all true science, 
and must be its foundation. 

" A plain and unadorned style," says 
Lord Bacon, "is the proper style for philo- 
sophy." Yet how far from being plain, — 
how studied in variety of phrase, — how re- 
dundant in poetic and metaphoric ornament, 
are the pages of a Stewart and Mackintosh ! 
Their pages seem to have been written under 



STEWART AND MACKINTOSH. Ill 

the impression that no effort of art should be 
spared to beguile the reader in the study of 
matters so dry and so laborious as ethics and 
metaphysics. But the result of this is to 
give the real student, who is carefully looking 
out for facts and principles, more trouble 
than can be estimated in finding the object of 
his search, — in separating the important 
from the trivial, — in testing, by the applica- 
tion of precise logic, the presence and 
quantity of meaning in a cloudy solution of 
verbiage. Their dissertations leave hardly 
any definite impression on the mind. Nor 
is this owing to any peculiar difficulty in 
coming at the facts which lay the foundation 
for principles, so much as to an erroneous 
view of the true method, or from a bad 
habit of philosophising, or perhaps to an in- 
disposition for that patient and cautious exa- 
mination of details, with that simplicity 
and exactness of phrase, which have few 
charms for any but the obscure devotees of 
truth. 

Metaphysical discussions are pre-eminently 
discussions about words. Ideas, feelings, 



112 STEWART AND MACKINTOSH. 

principles, faculties, powers, affections, ma- 
terialism, spiritualism, necessit}^, free-will, 
cu7n multis aliis, occur so constantly in these 
discussions, with so Httle exactness of mean- 
ing, and such variety of indistinct associated 
notions, that the student is still, like Milton's 
angels, " in wandering mazes lost." 

The dissertations of Stewart and Mackin- 
tosh, prefixed to the Encyclopaedia Britannica, 
enjoy a high degree of reputation. They are 
spoken of by critics of eminence as inimitable 
and invaluable. Reviews of great authority 
delight in recording the highest estimation of 
their merit. They are unquestionably use- 
ful and pleasing ; but of what are they the 
history ? If we compare them with the ac- 
companying dissertation by Playfair, on the 
History of Natural Philosophy, they must 
sink in the comparison, inasmuch as they 
fail to give that systematic view of real 
though gradual additions to human know- 
ledge, — of discoveries, with the authors, and 
times of discovery, which can alone properly 
constitute a history of any science or phi- 
losophy. They do not fasten upon im- 



Stewart's estimate of locke. 113 

portant epochs of improvement in the mode 
of studying and treating the philosophy oi 
mind. They do not give the details of writers 
and their systems, in the clear and exact 
manner most essential to the reader, nor sum 
up their respective merits and defects by the 
application of any well-stated, established, and 
acknowledged principles or criteria of excel- 
lence. 

Mr. Stewart makes the chief merit of 
Locke to consist in the inquiring spirit 
with which he imbues his reader. Speaking' 
of the general effect of Locke's discussions, 
in preparing the thinking part of his readers 
to a degree till then unknown, for the un- 
shackled use of their own reason, he says, 
"This has always appeared to me the most 
characteristical feature of Locke's Essay; and 
that to which it is chiefly indebted for its 
immense influence on the philosophy of the 
eighteenth century." The Essay of Locke 
is certainly the production of a profoundly 
thoughtful mind ; and the diligent reader, 
called into sympathy with such a mind, must 
partake, and imbibe in some degree, the spirit 



114 



of the great and good philosopher. But 
surely the merits of Locke, as the founder of 
a school of metaphysicians, are of a much 
higher order than such an estimate of his 
chief characteristics implies. Locke not only 
prepares the reader to reason, but prepares 
him to reason well ; to know himself, and 
man, and nature, better. It is difficult for 
us in these times to conceive the clouds and 
mists which the Essay on the Human Under- 
standing dissipated, and how it illuminated 
the whole horizon of human thought. Cud- 
worth's Intellectual System is a bock which 
sets the reader upon using his own reason ; 
and it requires a vastly superior stock of eru- 
dition to weigh and consider its contents. But 
Cudworth's learning oppresses the mind with 
its weight, and lies upon the intellect like a 
London fog upon the chest ; whereas Locke 
clears and braces the understanding. He 
has laid the broad and deep basis for a true 
intellectual system in his grand division of 
Sensation and Reflection ; the world without 
us, and the world within. His second chap- 
ter, on the Origin of our Ideas, and his last 



115 



chapter, on Knowledge, Reason, Faith, and 
Judgement, are full of rich, just, and invalua- 
ble matter. In the 28th chapter of the 2nd 
book he has given, incidentally, an analysis of 
moral laws, which must be in part followed by 
every other inquirer into ethics and jurispru- 
dence ; and the few words of Mr. Austin (in 
a note, p. 174 of his work on Jurisprudence) 
are far more worthy of the subject than all 
Mr. Stewart's elaborate though elegant 
phraseology. " Allowing for defects, which 
were nearly inevitable, his analysis is stri- 
kingly accurate. It evinces that matchless 
power of precise and just thinking, with that 
rehgious regard for general utility and truth, 
which marked the incomparable man who 
emancipated human reason from the yoke of 
mystery and jargon. And from this, his in- 
cidental excursions into the field of law and 
morality, and from other passages of his 
essay wherein he touches upon them, we 
may infer the important services which he 
would have rendered to the science of ethics 
if, complying with the instances of Molyneux, 
he had examined the subject exactly." 

Mr. Stewart's estimate of the defects 



116 LOCKE. 

and excellencies of Locke can never sa- 
tisfy the deep and cautious investigator of 
the constitution and powers of the human 
understanding ; and his everlasting, untiring 
intimations that the principles of Locke's 
Essay, if carried out, or not guarded by 
Locke's admissions, lead to the scepticism of 
Hume, are beyond measure unsatisfactory 
and wearisome to those who neither see nor 
understand the consequence, and who find it 
difficult to make out, at least from Mr. 
Stewart's statements, in what the scepticism 
of Hume precisely consisted, or how far it 
extended, save in the celebrated Essay on 
Miracles, where the uniformity of nature, 
or our own experience of it, is contrast- 
ed, in a loose manner, with the variable- 
ness of human testimon}^ and perhaps in a 
general want of principles and convictions in 
all the objects of metaphysical inquiry. 
There are two very different states of mind 
often confounded under the term scepti- 
cism ; one is that of an inquirer, who is de- 
sirous of placing his own meaning, and the 
meaning of others, before him in the clear- 
est and most satisfactory light, and pro- 



MACKINTOSH. 117 

portioning the degree of his assent to the 
strength of the evidence ; the other, that of 
a man who concludes positively against cer- 
tain opinions and propositions, as having no 
foundation in truth, merely because he can- 
not keep steadily before him the chain of 
arg-uments or evidence by which they are 
supported. The former is allied to what 
Hartley calls a " rehgious scepticism,'"' which 
receives nothing rashly, and is always pre- 
pared and seeking for evidence, with a view 
of concluding wisely and rationally. The lat- 
ter is less allied to scepticism than to dog- 
matism, and becomes dogmatism when a 
man, losing sight of the degrees of evidence 
by which the opinions he controverts are sup- 
ported, thinks to bear them down by the 
strength, rather than the justness, of his 
assertions. It is that of a man who is not 
seeking for truth, but doubts its existence ; 
who is in the habit of questioning every 
thing, and believing nothing. 

But if Mr. Stewart's work be open to cri- 
ticism, what shall we say of Sir James Mack- 
intosh, wdiose Dissertation excites the prin- 



118 MILL ON MACKINTOSH. 

cipal reviews to rapture, and which Mr. 
Whewell undertakes to preface and to edit ? 
That it is well — very well — worth perusal, may 
be asserted, partly because so little is written 
and so little is read with becoming care in 
such departments of human inquiry. But that 
the more it is read and examined, the less 
satisfactory it will appear, may also be assert- 
ed ; and he who does not choose to rest his 
justification of this assertion upon any ela- 
borate showing of his own, may shelter him- 
self behind Mill's Fragment on Mackintosh, 
and ask. Where is the answer to that book ? 
He may ask why the laudatory reviewers never 
allude to its existence, and why, since it con- 
tains so vigorous, hearty, and relentless an at- 
tack upon the merits of Sir James, there is no 
attempt to protect his fame, to meet and rebut 
its arguments. Surely Mr. Mill was no puny 
adversary — no mere fly, that stings but im- 
pedes not the noble racer in his dazzling 
course. The peaceful student of nature and 
of truth may not admire the somewhat bear- 
ish style in which Mr. Mill shakes and tears 
his prey ; and with his Fragment strews the 



MILL ON MACKINTOSH. 119 

ground with fragments of Sir James's mangled 
Dissertation. He may feel that the arrow from 
Mr. Mill's quiver, when he touches upon Sir 
James's " dandy philosophy" and lack-a- 
daisical style, has been dipped too cruelly in 
poison, tinged too darkly with the " gall of 
bitterness." Yet while his feelings of pity 
and tenderness are roused, and he cries out 
for mercy ! mercy ! on Sir James, it is a cry 
for mercy rather than for justice. The soft 
heart wishes to spare, and not to trample on 
the fallen. But Mr. Mill, roused by the ap- 
parent, though as he calls it affected candour, 
with real injustice to Mr. Bentham, descends 
like a second Achilles to avenge his friend, 
and drags his victim thrice round the field 
of his defeat. 

Seriously, he convicts Sir James over and 
over again of those superficial statements, of 
that vagueness and confusedness of lan- 
guage, which have hitherto been, and which 
continue to be, the great impediments to 
sound progress in the philosophy of mind. 
Yet, to do Sir James some justice, he is free 
from that aversion to acknowledge the full 



120 HARTLEY. 

extent and importance of the ''law of asso- 
ciation," which is characteristic of many, I 
believe most, of the Scotch metaphysical 
writers. He shares not in Dugald Stewart's 
antipathy to Hartley. He is the first meta- 
physical writer of considerable repute of late 
years, as far as 1 know, wdio seems to be 
somewhat duly aware of the merit and im- 
portance of the Hartleian theoiy and prin- 
ciples. He is wrong when he affects to see 
deeper into the moral constitution of man 
than others who have gone before him. He 
is convicted of error and absurdity in sup- 
posing that those of whom he was writing 
had overlooked a distinction which he had 
sagacity to discover between thoughts and 
feehngs, perceptions and emotions. And even 
while he admits that Hartley vras a great 
philosopher and a good man, there is hardly 
a sentence upon the nature of Hartley's sys- 
tem which conveys to the reader a distinct 
idea of it, — of its apprehended merits or 
defects. "Nothing," says Sir James, "more 
evidently points out the distinction of the 
Hartleian svstem from all svstems called self- 



HARTLEY. 121 

ish, not to say its superiority in respect to dis- 
interestedness over all moral systems before 
Butler and Hutcheson, than that excellent 
part of it which relates to the Rule of Life." 
Now the Rule of Life contains the whole 
moral system of Hartley, and is a deduction 
from the inquiry which Hartley instituted 
into the frame and constitution of man. 
And in the whole compass of human philo- 
sophy does any thing surpass that rule of 
life ? But for Butler and Hutcheson, it is 
evident that Hartley would have been the 
favourite philosopher of Mackintosh. Now 
since, according to his own showing, Butler 
and Hutcheson have really no system, but 
merely assert that human nature has certain 
characteristics, which no one ever denied, 
and which the philosophers of the Hartleian 
school have resolved, or tried to resolve, 
into the true elementary principles, we may 
perhaps, notwithstanding Mr. Mill's caustic 
animadversions, admit that the Dissertation 
of Sir James Mackintosh will, after all, help 
the student of mental philosophy to the 
quarter where hght is to be found. But he 

G 



122 PRESENT STATE 

must coatirme to be a student, and verify 
statements and weigh opinions for himself. 

It ma^T- appear unwarrantable to some 
readers that I should have lapsed from an 
inquiry into the nature of mathematical rea- 
soning and its application to science, sud- 
denly to a criticism upon the Dissertations 
of Stewart and Mackintosh. But the reason 
is obvious : these Dissertations appear to 
be the most popular, and so far the most 
important, treatises published of late upon 
the mental and moral constitution of man, 
which is doubtless the noblest of all human 
objects of inquiry and contemplation. I 
see or fancy that these Dissertations, both 
admitting and lamenting the ambiguity of 
language as the greatest obstacle in the way 
of the moral philosopher, are utterly defi-. 
cient in earnest and successful efforts to 
grapple with the difficulty and overcome it ; 
nay, increase rather than diminish the evil. 
True it is, that while we confine ourselves to 
that which we can rigidly and logically prove, 
in the present state of ethical science, 
we shall go but a little wa}", and we shall 



OF ETHICAL SCIENCE. 123 

soon come to all the conclusions which can 
be safely concluded, whether gathered from 
our most comprehensive inductions or flow- 
ing from the nature of our conceptions and 
terms. 



How little can be known ! 



This is tlie wise man's sigh : how far we err ! 
This is the good man's not unfrequent pang." 

The Excursion. 

But it is better to make sure of a little than 
to darken counsel by words without know- 
ledge ; it is better to build a small house fit 
and furnished to dwell in, than ruin our 
fortunes by undertaking a palace which we 
can neither finish nor inhabit ; and let us 
remember that philosophy is, after all, only 
strong sense, or the combined result of 
larger observation and deeper reflection in a 
little better dress than that of ordinary 
life. When a writer or reader sits down to 
philosophize, or to logical composition, he 
should consider first what it is he proposes 
to prove ; and secondly, by what means 
he proposes to prove it ; and he will soon 
find that he must make the nature and use 
g2 



124 LAPLACE ON PROBABILITIES. 

of language his principal study, as it is the 
main instrument of thought and communi- 
cation ; and in the proper use of it, consists 
the chief difference between the strong and 
the weak mind. 

" Scribencli recte sapere est et principium et fons." 

In considering the application of mathe- 
matical reasoning to moral and metaphysical 
subjects, it would be unpardonable to over- 
look the Philosophical Essay of Laplace, on 
Probabilities, reviewed by Playfair in the 
tY\"enty-third volume of the Edinburgh Review. 
Laplace has touched upon the applicability 
of the calculus to moral sciences in the fol- 
lowing order of topics : — First, the proba- 
bilities of testimony ; second, the votes and 
decisions of assemblies ; third, the judgements 
of legal tribunals ; fourth, tables of mortality, 
the mean duration of life, of marriages, and 
any social relations ; fifth, illusions in the 
estimate of probabilities. 

Although many just and some curious and 
novel observations are made by Laplace 
u.nder these heads, vet their general efiect is 



LAPLACE ON PROBABILITIES. 125 

to show the wide difference which exists be- 
tween those events which, by a strict and 
clear resemblance, can be classed together 
and subjected to numerical calculation, and 
those which are of more complicated charac- 
ter, have a less perfect -analogy, differ from 
each other by many circumstances peculiar 
to each case, — diversities growing out of the 
general mental, and moral constitution of 
man, and which cannot, therefore, be so 
legitimately classed, nor so successfully 
brought within the category of number. 
It is indeed somewhat singular, that La- 
place should have placed tables of mor- 
tality and the duration of life, and the re- 
lative proportion of births, marriages, and 
deaths, to population, — events of such uni- 
formity of character, and so easily and na- 
turally classed, — in connexion with subjects 
so wide and general as the nature of testi- 
mony, the decisions of popular assemblies 
and legal tribunals, and the illusions of ima- 
gination, — subjects which he has touched 
but slightly. Statistical calculations ought 
not indeed to be overlooked by the moralist, 



126 PLAYFAIR ON LAPLACE. 

since they furnish that extended view of 
the phenomena of Hfe and of the moral 
world, which is essential to the deduction of 
sound moral rules and principles, and af- 
fords great help to the interpretation of the 
laws and ends of Divine Providence in the 
administration of the world. Any collection 
of analogous phenomena in the physical 
world is of importance in the moral, since 
such collections lay the foundation of general 
rules, and supply the exact knowledge which 
is of the greatest use in the conduct of life. 
But in collecting analogies we must not 
overlook differences, and be duly cautious 
in the apphcation of general maxims, which 
are apt to mislead as well as to instruct. 
Playfair has made some just remarks on the 
vagueness of Laplace's language and views 
on the subject of testimony ; and it is easy 
to conceive that Laplace's devotedness to 
the exact sciences may have disposed him to 
an unreasonable mistrust and disregard of all 
conclusions of the pure intellect, drawn from 
the phenomena of history and the moral 
world. But when Playfair intimates that no 



LAPLACE ON ASSOCIATION. 127 

conclusion founded on the application of the 
calculus to moral probabilities should be al- 
lowed to interfere luith the truths of religion, 
he seems to imply that such an interference 
may be anticipated, and rationally feared, and 
to forget that, to the philosophic mind, truth 
in all departments must be ever consistent 
with itself; that though the atoms of the 
universe may be weighed and measured, and 
every seeming accident shall be reduced to 
order and to rule, the reasons for adoring the 
Creator, and trusting in him and obeying 
him, and for loving our neighbour as our- 
selves, will only be proportionably multi- 
plied. 

It is to me truly interesting to observe La- 
place perceiving and recording the impor- 
tance of that principle of association into 
which Hartley, foUov/ed by Mill, resolves all 
reasoning. " The most fertile of all the 
principles of psychology," (Laplace, p. 224, 
of the 8vo. edition of his Essay,) " is that of 
the association of all things which have had 
in the sensorium a simultaneous or regularly 
successive existence ; an association whereby 



128 LAPLACE ON ASSOCIATION. 

the return of one calls up all the others con- 
nected with it ; the objects which we have 
formerly seen awaken the traces of things 
which in the first view were associated with 
them. These traces call up in the same 
manner those of other objects, and so on in 
succession, so that by means of one thing 
presented to the mind, we can recal an infi- 
nity of others, and rest our attention upon 
whatever we wish to consider. To this 
principle the employment of signs and lan- 
guage for recalling sensations and ideas be- 
longs ; it accounts for the formation and 
analysis of complex, abstract, and general 
ideas, and for reasoning. Many philosophers 
have well developed this principle, which up 
to the present time constitutes the real part 
of metaphysics." " I have endeavoured to 
show in these papers," says Hartley, at the 
conclusion of his 99th proposition, " that 
all reasoning, as well as affection, is the mere 
result of association." 

Some further extracts from Laplace might 
perhaps be given with interest to the reader, 
but it does not appear to me that Laplace 



Austin's jurisprudence. 129 

adds any thing to the ingenious observations 
and speculations of Hartley, in his chapter 
on the Deduction of Rules for the ascertain- 
ment of Truth, and advancement of Know- 
ledge, from the mathematical Methods of con- 
sidering Quantity. Mr. Dugald Stewart, in 
the Preliminary Dissertation to his Philoso- 
phical Essays, was pleased to sneer at some 
of the most refined and natural conclusions 
of that great and good philosopher, Hartley, 
in the chapter referred to, and to consider 
them as evidence of the unsoundness of his 
understanding. But Mr. Stewart is very 
often unfortunate in his selection of the sen- 
timents of the greatest English metaphysi- 
cians, which he marks for peculiar reproba- 
tion and contempt. 

I have now brought together, and laid 
before the reader, some thoughts which have 
occurred to me when reading the more po- 
pular scientific publications of the day. I am 
not without hope of awakening the attention 
of a few, into whose hands these pages may 
fall, to the contents and merits of authors 
whose works they may not already suffi- 



130 Austin's jurisprudence. 

ciently appreciate, but which it is good " to 
feed upon, as insects on a leaf, till the whole 
heart be coloured vv^itli the fibre." The rea- 
der w^ho wishes to see one of the best examples 
which our times have afforded of strictly lo- 
gical or demonstrative reasoning applied to 
moral and metaphysical subjects, will do vvell 
to study the w^ork of Mr. John Austin, if he is 
not already acquainted with it, entitled, " The 
Province of Jurisprudence Determined." 
being the substance of lectures delivered 
at the University of London as Professor of 
Jurisprudence. Contrary to the practice of 
those numerous writers who confound that 
of which they are treating with every thing 
else with which it can, by possibility, be con- 
, founded, Mr. Austin carefully distinguishes 
his subject from every thing with which it is, 
howsoever remotely, related. He is one of 
the few writers who does not fear to repeat 
himself, so much as to be misunderstood by 
his reader. He w^ould rather give you a whole 
sentence tAvo or three times over, than intro- 
duce a pronoun whose antecedent might be 
doubtful or nonexistent. He seeks not, bv 



Austin's jurisprudence. 131 

showy phrase, to impose on the understanding 
of his reader, or to smooth over with received 
verbiage the difficulties of his subject and the 
absence of thought ; and if he sometimes 
wearies with the repetition of a lengthened 
phrase, for which a convenient abbreviation 
might have been advantageously adopted, we 
could forgive much more than this to a writer 
who feels so acutely the audacity of the para- 
dox, " that men really should think distinctly 
and speak with a meaning." As it bears very 
closely, indeed, upon the topic discussed in 
these pages, and as the matter is itself of the 
highest importance in connexion with ethical 
inquiries, I shall not scruple to close this 
essay with the following extract from Mr. 
Austin's work, feeling, as I do, its fullness 
of truth : — 

" If there were a reading public nume- 
rous, discerning, and impartial, the science 
of ethics, and all the various sciences which 
are nearly related to ethics, would advance 
with unexampled rapidity. 

" By the hope of obtaining the approba- 
tion which it would bestow upon genuine 



132 Austin's jurisprudence. 

merit, writers would be incited to the patient 
research and reflection which are not less re- 
quisite to the improvement of ethical than 
to the advancement of mathematical science. 

" Slight and incoherent thinking v/ould be 
received with general contempt, though it 
were cased in polished periods, studded Vvith 
brilliant metaphors. Ethics would be con- 
sidered by readers, and therefore treated by 
writers, as the matter or subject of a sci- 
ence ; as a subject for persevering and ac- 
curate investigation, and not as a theme for 
childish and babbling rhetoric. 

" This general demand for truth, (though 
it were clothed in homely guise,) and this 
general contempt of falsehood and nonsense, 
(though they were decked with rhetorical 
graces,) would improve the method and the 
style of inquiries into ethics, and into the 
various sciences which are nearly related to 
ethics. The writers would attend to the 
suggestions of Hobbes and of Locke, and 
would imitate the method so successfully 
pursued by geometers ; though such is the 
variety of the premises which some of their 



Austin's jurisprudence. - 133 

inquiries involve, and such are the com- 
plexity and ambiguity of some of the terms, 
that they would often fall short of the per- 
fect exactness and coherency which the few- 
ness of his premises, and the simphcity and 
definiteness of his expressions, enable the 
geometer to reach. But though they would 
often fall short of geometrical exactness and 
coherency, they might always approach, and 
would often attain, to them : they would 
acquire the art and the habit of defining 
their leading terms ; of steadily adhering to 
the meanings announced by the definitions ; 
of carefally examining, and distinctly stating, 
their premises ; and of deducing the con- 
sequences of their premise's with logical 
rigour. Without rejecting embellishments 
which might happen to fall in their way, the 
only excellencies of style for which they 
would seek are precision, clearness, and con- 
ciseness ; the first being absolutely requisite 
to the successful prosecution of inquirj^ 
whilst the others enable the reader to seize 
the m.eaning with certainty, and spare him 
unnecessary fatigue. 



134 Austin's jurisprudence. 

" And what is equally important, the pro- 
tection afforded by this public to diligent and 
honest writers, would inspire into writers 
upon ethics, and upon the nearly-related 
sciences, the spirit of dispassionate inquiry, 
the indifferency or impartiality in the pursuit 
of truth, which is just as requisite to the 
detection of truth as continued and close 
attention, or sincerity and simplicity of pur- 
pose. Relying on the discernment and the 
justice of a numerous and powerful public, 
shielded by its countenance from the shafts 
of the hypocrite and the bigot, — ^indifferent 
to the idle whistling of that harmless storm, 
they would scrutinize established institutions 
and current or received opinions fearlessly, 
but coolly, with the freedom which is de- 
manded by general utility, but without the 
antipathy which is begotten by the dread of 
persecution, and wliich is scarcely less ad- 
verse than ' the love of things ancient' to 
the rapid advancement of science. 

" This patience in investigation, this dis- 
tinctness and accuracy of method, this free- 
dom and ' indifferency ' in the pursuit of the 



Austin's jurisprudence. 135 

useful and the true, would thoroughly dispel 
the obscurity by which the science is clouded, 
and would clear it from most of its uncer- 
tainties. The wish, the hope, the prediction 
of Mr. Locke, would, in time, be accom- 
plished, and ' ethics v\ould rank with the 
sciences which are capable of demonstration.' 
The adepts in ethical as well as in mathema- 
tical science would commonly agree in their 
results ; and as the jar of their conclusions 
gradually subsided, a body of doctrine and 
authority to which the multitude might trust 
would emerge from the existing chaos." 



FINIS. 



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